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## G = C22.52(S3×Q8)  order 192 = 26·3

### 15th central stem extension by C22 of S3×Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C22.52(S3×Q8)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — C22.52(S3×Q8)
 Lower central C3 — C22×C6 — C22.52(S3×Q8)
 Upper central C1 — C23 — C22×Q8

Generators and relations for C22.52(S3×Q8)
G = < a,b,c,d,e,f | a2=b2=c3=e4=1, d2=b, f2=e2, ab=ba, ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, dcd-1=c-1, ce=ec, cf=fc, fef-1=e-1 >

Subgroups: 408 in 182 conjugacy classes, 69 normal (12 characteristic)
C1, C2, C2 [×6], C3, C4 [×13], C22, C22 [×6], C6, C6 [×6], C2×C4 [×6], C2×C4 [×21], Q8 [×8], C23, Dic3 [×7], C12 [×6], C2×C6, C2×C6 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×6], C2×Dic3 [×6], C2×Dic3 [×9], C2×C12 [×6], C2×C12 [×6], C3×Q8 [×8], C22×C6, C2.C42 [×3], C2×C4⋊C4 [×3], C22×Q8, Dic3⋊C4 [×6], C22×Dic3, C22×Dic3 [×3], C22×C12 [×3], C6×Q8 [×6], C23.78C23, C6.C42 [×3], C2×Dic3⋊C4 [×3], Q8×C2×C6, C22.52(S3×Q8)
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], Q8 [×6], C23, D6 [×3], C2×D4 [×3], C2×Q8 [×3], C4○D4, C3⋊D4 [×6], C22×S3, C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], S3×Q8 [×3], Q83S3, C2×C3⋊D4 [×3], C23.78C23, Dic3⋊Q8 [×3], D63Q8 [×3], C244S3, C22.52(S3×Q8)

Smallest permutation representation of C22.52(S3×Q8)
Regular action on 192 points
Generators in S192
(1 39)(2 40)(3 37)(4 38)(5 61)(6 62)(7 63)(8 64)(9 126)(10 127)(11 128)(12 125)(13 118)(14 119)(15 120)(16 117)(17 124)(18 121)(19 122)(20 123)(21 138)(22 139)(23 140)(24 137)(25 130)(26 131)(27 132)(28 129)(29 136)(30 133)(31 134)(32 135)(33 105)(34 106)(35 107)(36 108)(41 54)(42 55)(43 56)(44 53)(45 99)(46 100)(47 97)(48 98)(49 184)(50 181)(51 182)(52 183)(57 176)(58 173)(59 174)(60 175)(65 149)(66 150)(67 151)(68 152)(69 188)(70 185)(71 186)(72 187)(73 143)(74 144)(75 141)(76 142)(77 159)(78 160)(79 157)(80 158)(81 153)(82 154)(83 155)(84 156)(85 161)(86 162)(87 163)(88 164)(89 171)(90 172)(91 169)(92 170)(93 165)(94 166)(95 167)(96 168)(101 177)(102 178)(103 179)(104 180)(109 146)(110 147)(111 148)(112 145)(113 189)(114 190)(115 191)(116 192)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)(65 67)(66 68)(69 71)(70 72)(73 75)(74 76)(77 79)(78 80)(81 83)(82 84)(85 87)(86 88)(89 91)(90 92)(93 95)(94 96)(97 99)(98 100)(101 103)(102 104)(105 107)(106 108)(109 111)(110 112)(113 115)(114 116)(117 119)(118 120)(121 123)(122 124)(125 127)(126 128)(129 131)(130 132)(133 135)(134 136)(137 139)(138 140)(141 143)(142 144)(145 147)(146 148)(149 151)(150 152)(153 155)(154 156)(157 159)(158 160)(161 163)(162 164)(165 167)(166 168)(169 171)(170 172)(173 175)(174 176)(177 179)(178 180)(181 183)(182 184)(185 187)(186 188)(189 191)(190 192)
(1 54 35)(2 36 55)(3 56 33)(4 34 53)(5 114 71)(6 72 115)(7 116 69)(8 70 113)(9 118 122)(10 123 119)(11 120 124)(12 121 117)(13 19 126)(14 127 20)(15 17 128)(16 125 18)(21 130 134)(22 135 131)(23 132 136)(24 133 129)(25 31 138)(26 139 32)(27 29 140)(28 137 30)(37 43 105)(38 106 44)(39 41 107)(40 108 42)(45 111 68)(46 65 112)(47 109 66)(48 67 110)(49 178 174)(50 175 179)(51 180 176)(52 173 177)(57 182 104)(58 101 183)(59 184 102)(60 103 181)(61 190 186)(62 187 191)(63 192 188)(64 185 189)(73 81 159)(74 160 82)(75 83 157)(76 158 84)(77 143 153)(78 154 144)(79 141 155)(80 156 142)(85 93 171)(86 172 94)(87 95 169)(88 170 96)(89 161 165)(90 166 162)(91 163 167)(92 168 164)(97 146 150)(98 151 147)(99 148 152)(100 149 145)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)(97 98 99 100)(101 102 103 104)(105 106 107 108)(109 110 111 112)(113 114 115 116)(117 118 119 120)(121 122 123 124)(125 126 127 128)(129 130 131 132)(133 134 135 136)(137 138 139 140)(141 142 143 144)(145 146 147 148)(149 150 151 152)(153 154 155 156)(157 158 159 160)(161 162 163 164)(165 166 167 168)(169 170 171 172)(173 174 175 176)(177 178 179 180)(181 182 183 184)(185 186 187 188)(189 190 191 192)
(1 187 175 109)(2 69 176 147)(3 185 173 111)(4 71 174 145)(5 49 100 34)(6 181 97 107)(7 51 98 36)(8 183 99 105)(9 21 77 89)(10 139 78 172)(11 23 79 91)(12 137 80 170)(13 25 73 85)(14 131 74 162)(15 27 75 87)(16 129 76 164)(17 29 83 95)(18 133 84 168)(19 31 81 93)(20 135 82 166)(22 160 90 127)(24 158 92 125)(26 144 86 119)(28 142 88 117)(30 156 96 121)(32 154 94 123)(33 64 52 45)(35 62 50 47)(37 70 58 148)(38 186 59 112)(39 72 60 146)(40 188 57 110)(41 115 103 150)(42 192 104 67)(43 113 101 152)(44 190 102 65)(46 106 61 184)(48 108 63 182)(53 114 178 149)(54 191 179 66)(55 116 180 151)(56 189 177 68)(118 130 143 161)(120 132 141 163)(122 134 153 165)(124 136 155 167)(126 138 159 171)(128 140 157 169)
(1 120 175 141)(2 119 176 144)(3 118 173 143)(4 117 174 142)(5 170 100 137)(6 169 97 140)(7 172 98 139)(8 171 99 138)(9 52 77 33)(10 51 78 36)(11 50 79 35)(12 49 80 34)(13 58 73 37)(14 57 74 40)(15 60 75 39)(16 59 76 38)(17 103 83 41)(18 102 84 44)(19 101 81 43)(20 104 82 42)(21 64 89 45)(22 63 90 48)(23 62 91 47)(24 61 92 46)(25 70 85 148)(26 69 86 147)(27 72 87 146)(28 71 88 145)(29 115 95 150)(30 114 96 149)(31 113 93 152)(32 116 94 151)(53 121 178 156)(54 124 179 155)(55 123 180 154)(56 122 177 153)(65 133 190 168)(66 136 191 167)(67 135 192 166)(68 134 189 165)(105 126 183 159)(106 125 184 158)(107 128 181 157)(108 127 182 160)(109 132 187 163)(110 131 188 162)(111 130 185 161)(112 129 186 164)

G:=sub<Sym(192)| (1,39)(2,40)(3,37)(4,38)(5,61)(6,62)(7,63)(8,64)(9,126)(10,127)(11,128)(12,125)(13,118)(14,119)(15,120)(16,117)(17,124)(18,121)(19,122)(20,123)(21,138)(22,139)(23,140)(24,137)(25,130)(26,131)(27,132)(28,129)(29,136)(30,133)(31,134)(32,135)(33,105)(34,106)(35,107)(36,108)(41,54)(42,55)(43,56)(44,53)(45,99)(46,100)(47,97)(48,98)(49,184)(50,181)(51,182)(52,183)(57,176)(58,173)(59,174)(60,175)(65,149)(66,150)(67,151)(68,152)(69,188)(70,185)(71,186)(72,187)(73,143)(74,144)(75,141)(76,142)(77,159)(78,160)(79,157)(80,158)(81,153)(82,154)(83,155)(84,156)(85,161)(86,162)(87,163)(88,164)(89,171)(90,172)(91,169)(92,170)(93,165)(94,166)(95,167)(96,168)(101,177)(102,178)(103,179)(104,180)(109,146)(110,147)(111,148)(112,145)(113,189)(114,190)(115,191)(116,192), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192), (1,54,35)(2,36,55)(3,56,33)(4,34,53)(5,114,71)(6,72,115)(7,116,69)(8,70,113)(9,118,122)(10,123,119)(11,120,124)(12,121,117)(13,19,126)(14,127,20)(15,17,128)(16,125,18)(21,130,134)(22,135,131)(23,132,136)(24,133,129)(25,31,138)(26,139,32)(27,29,140)(28,137,30)(37,43,105)(38,106,44)(39,41,107)(40,108,42)(45,111,68)(46,65,112)(47,109,66)(48,67,110)(49,178,174)(50,175,179)(51,180,176)(52,173,177)(57,182,104)(58,101,183)(59,184,102)(60,103,181)(61,190,186)(62,187,191)(63,192,188)(64,185,189)(73,81,159)(74,160,82)(75,83,157)(76,158,84)(77,143,153)(78,154,144)(79,141,155)(80,156,142)(85,93,171)(86,172,94)(87,95,169)(88,170,96)(89,161,165)(90,166,162)(91,163,167)(92,168,164)(97,146,150)(98,151,147)(99,148,152)(100,149,145), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144)(145,146,147,148)(149,150,151,152)(153,154,155,156)(157,158,159,160)(161,162,163,164)(165,166,167,168)(169,170,171,172)(173,174,175,176)(177,178,179,180)(181,182,183,184)(185,186,187,188)(189,190,191,192), (1,187,175,109)(2,69,176,147)(3,185,173,111)(4,71,174,145)(5,49,100,34)(6,181,97,107)(7,51,98,36)(8,183,99,105)(9,21,77,89)(10,139,78,172)(11,23,79,91)(12,137,80,170)(13,25,73,85)(14,131,74,162)(15,27,75,87)(16,129,76,164)(17,29,83,95)(18,133,84,168)(19,31,81,93)(20,135,82,166)(22,160,90,127)(24,158,92,125)(26,144,86,119)(28,142,88,117)(30,156,96,121)(32,154,94,123)(33,64,52,45)(35,62,50,47)(37,70,58,148)(38,186,59,112)(39,72,60,146)(40,188,57,110)(41,115,103,150)(42,192,104,67)(43,113,101,152)(44,190,102,65)(46,106,61,184)(48,108,63,182)(53,114,178,149)(54,191,179,66)(55,116,180,151)(56,189,177,68)(118,130,143,161)(120,132,141,163)(122,134,153,165)(124,136,155,167)(126,138,159,171)(128,140,157,169), (1,120,175,141)(2,119,176,144)(3,118,173,143)(4,117,174,142)(5,170,100,137)(6,169,97,140)(7,172,98,139)(8,171,99,138)(9,52,77,33)(10,51,78,36)(11,50,79,35)(12,49,80,34)(13,58,73,37)(14,57,74,40)(15,60,75,39)(16,59,76,38)(17,103,83,41)(18,102,84,44)(19,101,81,43)(20,104,82,42)(21,64,89,45)(22,63,90,48)(23,62,91,47)(24,61,92,46)(25,70,85,148)(26,69,86,147)(27,72,87,146)(28,71,88,145)(29,115,95,150)(30,114,96,149)(31,113,93,152)(32,116,94,151)(53,121,178,156)(54,124,179,155)(55,123,180,154)(56,122,177,153)(65,133,190,168)(66,136,191,167)(67,135,192,166)(68,134,189,165)(105,126,183,159)(106,125,184,158)(107,128,181,157)(108,127,182,160)(109,132,187,163)(110,131,188,162)(111,130,185,161)(112,129,186,164)>;

G:=Group( (1,39)(2,40)(3,37)(4,38)(5,61)(6,62)(7,63)(8,64)(9,126)(10,127)(11,128)(12,125)(13,118)(14,119)(15,120)(16,117)(17,124)(18,121)(19,122)(20,123)(21,138)(22,139)(23,140)(24,137)(25,130)(26,131)(27,132)(28,129)(29,136)(30,133)(31,134)(32,135)(33,105)(34,106)(35,107)(36,108)(41,54)(42,55)(43,56)(44,53)(45,99)(46,100)(47,97)(48,98)(49,184)(50,181)(51,182)(52,183)(57,176)(58,173)(59,174)(60,175)(65,149)(66,150)(67,151)(68,152)(69,188)(70,185)(71,186)(72,187)(73,143)(74,144)(75,141)(76,142)(77,159)(78,160)(79,157)(80,158)(81,153)(82,154)(83,155)(84,156)(85,161)(86,162)(87,163)(88,164)(89,171)(90,172)(91,169)(92,170)(93,165)(94,166)(95,167)(96,168)(101,177)(102,178)(103,179)(104,180)(109,146)(110,147)(111,148)(112,145)(113,189)(114,190)(115,191)(116,192), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64)(65,67)(66,68)(69,71)(70,72)(73,75)(74,76)(77,79)(78,80)(81,83)(82,84)(85,87)(86,88)(89,91)(90,92)(93,95)(94,96)(97,99)(98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192), (1,54,35)(2,36,55)(3,56,33)(4,34,53)(5,114,71)(6,72,115)(7,116,69)(8,70,113)(9,118,122)(10,123,119)(11,120,124)(12,121,117)(13,19,126)(14,127,20)(15,17,128)(16,125,18)(21,130,134)(22,135,131)(23,132,136)(24,133,129)(25,31,138)(26,139,32)(27,29,140)(28,137,30)(37,43,105)(38,106,44)(39,41,107)(40,108,42)(45,111,68)(46,65,112)(47,109,66)(48,67,110)(49,178,174)(50,175,179)(51,180,176)(52,173,177)(57,182,104)(58,101,183)(59,184,102)(60,103,181)(61,190,186)(62,187,191)(63,192,188)(64,185,189)(73,81,159)(74,160,82)(75,83,157)(76,158,84)(77,143,153)(78,154,144)(79,141,155)(80,156,142)(85,93,171)(86,172,94)(87,95,169)(88,170,96)(89,161,165)(90,166,162)(91,163,167)(92,168,164)(97,146,150)(98,151,147)(99,148,152)(100,149,145), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128)(129,130,131,132)(133,134,135,136)(137,138,139,140)(141,142,143,144)(145,146,147,148)(149,150,151,152)(153,154,155,156)(157,158,159,160)(161,162,163,164)(165,166,167,168)(169,170,171,172)(173,174,175,176)(177,178,179,180)(181,182,183,184)(185,186,187,188)(189,190,191,192), (1,187,175,109)(2,69,176,147)(3,185,173,111)(4,71,174,145)(5,49,100,34)(6,181,97,107)(7,51,98,36)(8,183,99,105)(9,21,77,89)(10,139,78,172)(11,23,79,91)(12,137,80,170)(13,25,73,85)(14,131,74,162)(15,27,75,87)(16,129,76,164)(17,29,83,95)(18,133,84,168)(19,31,81,93)(20,135,82,166)(22,160,90,127)(24,158,92,125)(26,144,86,119)(28,142,88,117)(30,156,96,121)(32,154,94,123)(33,64,52,45)(35,62,50,47)(37,70,58,148)(38,186,59,112)(39,72,60,146)(40,188,57,110)(41,115,103,150)(42,192,104,67)(43,113,101,152)(44,190,102,65)(46,106,61,184)(48,108,63,182)(53,114,178,149)(54,191,179,66)(55,116,180,151)(56,189,177,68)(118,130,143,161)(120,132,141,163)(122,134,153,165)(124,136,155,167)(126,138,159,171)(128,140,157,169), (1,120,175,141)(2,119,176,144)(3,118,173,143)(4,117,174,142)(5,170,100,137)(6,169,97,140)(7,172,98,139)(8,171,99,138)(9,52,77,33)(10,51,78,36)(11,50,79,35)(12,49,80,34)(13,58,73,37)(14,57,74,40)(15,60,75,39)(16,59,76,38)(17,103,83,41)(18,102,84,44)(19,101,81,43)(20,104,82,42)(21,64,89,45)(22,63,90,48)(23,62,91,47)(24,61,92,46)(25,70,85,148)(26,69,86,147)(27,72,87,146)(28,71,88,145)(29,115,95,150)(30,114,96,149)(31,113,93,152)(32,116,94,151)(53,121,178,156)(54,124,179,155)(55,123,180,154)(56,122,177,153)(65,133,190,168)(66,136,191,167)(67,135,192,166)(68,134,189,165)(105,126,183,159)(106,125,184,158)(107,128,181,157)(108,127,182,160)(109,132,187,163)(110,131,188,162)(111,130,185,161)(112,129,186,164) );

G=PermutationGroup([(1,39),(2,40),(3,37),(4,38),(5,61),(6,62),(7,63),(8,64),(9,126),(10,127),(11,128),(12,125),(13,118),(14,119),(15,120),(16,117),(17,124),(18,121),(19,122),(20,123),(21,138),(22,139),(23,140),(24,137),(25,130),(26,131),(27,132),(28,129),(29,136),(30,133),(31,134),(32,135),(33,105),(34,106),(35,107),(36,108),(41,54),(42,55),(43,56),(44,53),(45,99),(46,100),(47,97),(48,98),(49,184),(50,181),(51,182),(52,183),(57,176),(58,173),(59,174),(60,175),(65,149),(66,150),(67,151),(68,152),(69,188),(70,185),(71,186),(72,187),(73,143),(74,144),(75,141),(76,142),(77,159),(78,160),(79,157),(80,158),(81,153),(82,154),(83,155),(84,156),(85,161),(86,162),(87,163),(88,164),(89,171),(90,172),(91,169),(92,170),(93,165),(94,166),(95,167),(96,168),(101,177),(102,178),(103,179),(104,180),(109,146),(110,147),(111,148),(112,145),(113,189),(114,190),(115,191),(116,192)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64),(65,67),(66,68),(69,71),(70,72),(73,75),(74,76),(77,79),(78,80),(81,83),(82,84),(85,87),(86,88),(89,91),(90,92),(93,95),(94,96),(97,99),(98,100),(101,103),(102,104),(105,107),(106,108),(109,111),(110,112),(113,115),(114,116),(117,119),(118,120),(121,123),(122,124),(125,127),(126,128),(129,131),(130,132),(133,135),(134,136),(137,139),(138,140),(141,143),(142,144),(145,147),(146,148),(149,151),(150,152),(153,155),(154,156),(157,159),(158,160),(161,163),(162,164),(165,167),(166,168),(169,171),(170,172),(173,175),(174,176),(177,179),(178,180),(181,183),(182,184),(185,187),(186,188),(189,191),(190,192)], [(1,54,35),(2,36,55),(3,56,33),(4,34,53),(5,114,71),(6,72,115),(7,116,69),(8,70,113),(9,118,122),(10,123,119),(11,120,124),(12,121,117),(13,19,126),(14,127,20),(15,17,128),(16,125,18),(21,130,134),(22,135,131),(23,132,136),(24,133,129),(25,31,138),(26,139,32),(27,29,140),(28,137,30),(37,43,105),(38,106,44),(39,41,107),(40,108,42),(45,111,68),(46,65,112),(47,109,66),(48,67,110),(49,178,174),(50,175,179),(51,180,176),(52,173,177),(57,182,104),(58,101,183),(59,184,102),(60,103,181),(61,190,186),(62,187,191),(63,192,188),(64,185,189),(73,81,159),(74,160,82),(75,83,157),(76,158,84),(77,143,153),(78,154,144),(79,141,155),(80,156,142),(85,93,171),(86,172,94),(87,95,169),(88,170,96),(89,161,165),(90,166,162),(91,163,167),(92,168,164),(97,146,150),(98,151,147),(99,148,152),(100,149,145)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96),(97,98,99,100),(101,102,103,104),(105,106,107,108),(109,110,111,112),(113,114,115,116),(117,118,119,120),(121,122,123,124),(125,126,127,128),(129,130,131,132),(133,134,135,136),(137,138,139,140),(141,142,143,144),(145,146,147,148),(149,150,151,152),(153,154,155,156),(157,158,159,160),(161,162,163,164),(165,166,167,168),(169,170,171,172),(173,174,175,176),(177,178,179,180),(181,182,183,184),(185,186,187,188),(189,190,191,192)], [(1,187,175,109),(2,69,176,147),(3,185,173,111),(4,71,174,145),(5,49,100,34),(6,181,97,107),(7,51,98,36),(8,183,99,105),(9,21,77,89),(10,139,78,172),(11,23,79,91),(12,137,80,170),(13,25,73,85),(14,131,74,162),(15,27,75,87),(16,129,76,164),(17,29,83,95),(18,133,84,168),(19,31,81,93),(20,135,82,166),(22,160,90,127),(24,158,92,125),(26,144,86,119),(28,142,88,117),(30,156,96,121),(32,154,94,123),(33,64,52,45),(35,62,50,47),(37,70,58,148),(38,186,59,112),(39,72,60,146),(40,188,57,110),(41,115,103,150),(42,192,104,67),(43,113,101,152),(44,190,102,65),(46,106,61,184),(48,108,63,182),(53,114,178,149),(54,191,179,66),(55,116,180,151),(56,189,177,68),(118,130,143,161),(120,132,141,163),(122,134,153,165),(124,136,155,167),(126,138,159,171),(128,140,157,169)], [(1,120,175,141),(2,119,176,144),(3,118,173,143),(4,117,174,142),(5,170,100,137),(6,169,97,140),(7,172,98,139),(8,171,99,138),(9,52,77,33),(10,51,78,36),(11,50,79,35),(12,49,80,34),(13,58,73,37),(14,57,74,40),(15,60,75,39),(16,59,76,38),(17,103,83,41),(18,102,84,44),(19,101,81,43),(20,104,82,42),(21,64,89,45),(22,63,90,48),(23,62,91,47),(24,61,92,46),(25,70,85,148),(26,69,86,147),(27,72,87,146),(28,71,88,145),(29,115,95,150),(30,114,96,149),(31,113,93,152),(32,116,94,151),(53,121,178,156),(54,124,179,155),(55,123,180,154),(56,122,177,153),(65,133,190,168),(66,136,191,167),(67,135,192,166),(68,134,189,165),(105,126,183,159),(106,125,184,158),(107,128,181,157),(108,127,182,160),(109,132,187,163),(110,131,188,162),(111,130,185,161),(112,129,186,164)])

42 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4F 4G ··· 4N 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + - + + - + image C1 C2 C2 C2 S3 Q8 D4 D6 C4○D4 C3⋊D4 S3×Q8 Q8⋊3S3 kernel C22.52(S3×Q8) C6.C42 C2×Dic3⋊C4 Q8×C2×C6 C22×Q8 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 3 3 1 1 6 6 3 2 12 3 1

Matrix representation of C22.52(S3×Q8) in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 3 0 0 0 0 6 3 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 0
,
 2 4 0 0 0 0 9 11 0 0 0 0 0 0 6 3 0 0 0 0 5 7 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 2 0 0 0 0 11 9 0 0 0 0 0 0 9 3 0 0 0 0 3 4

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,6,0,0,0,0,3,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,6,5,0,0,0,0,3,7,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,11,0,0,0,0,2,9,0,0,0,0,0,0,9,3,0,0,0,0,3,4] >;

C22.52(S3×Q8) in GAP, Magma, Sage, TeX

C_2^2._{52}(S_3\times Q_8)
% in TeX

G:=Group("C2^2.52(S3xQ8)");
// GroupNames label

G:=SmallGroup(192,789);
// by ID

G=gap.SmallGroup(192,789);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,254,387,184,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=e^4=1,d^2=b,f^2=e^2,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=c^-1,c*e=e*c,c*f=f*c,f*e*f^-1=e^-1>;
// generators/relations

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