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G = C22×C3⋊Q16  order 192 = 26·3

Direct product of C22 and C3⋊Q16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×C3⋊Q16, C12.32C24, Dic6.28C23, (C2×C6)⋊9Q16, C6⋊3(C2×Q16), C3⋊3(C22×Q16), C3⋊C8.29C23, (C2×C12).212D4, C12.256(C2×D4), C4.32(S3×C23), (C2×Q8).209D6, (C22×C6).211D4, C6.151(C22×D4), (C22×C4).398D6, (C22×Q8).13S3, (C3×Q8).21C23, Q8.31(C22×S3), (C2×C12).549C23, (C6×Q8).230C22, C23.114(C3⋊D4), (C22×Dic6).18C2, (C22×C12).281C22, (C2×Dic6).306C22, (Q8×C2×C6).6C2, C4.26(C2×C3⋊D4), (C2×C6).586(C2×D4), (C22×C3⋊C8).14C2, (C2×C3⋊C8).287C22, C2.24(C22×C3⋊D4), (C2×C4).155(C3⋊D4), (C2×C4).630(C22×S3), C22.114(C2×C3⋊D4), SmallGroup(192,1368)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C22×C3⋊Q16
Chief series
C1 — C3 — C6 — C12 — Dic6 — C2×Dic6 — C22×Dic6 — C22×C3⋊Q16
Lower central
C3 — C6 — C12 — C22×C3⋊Q16

Subgroups: 520 in 258 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C4 [×8], C22 [×7], C6, C6 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×4], Q8 [×16], C23, Dic3 [×4], C12, C12 [×3], C12 [×4], C2×C6 [×7], C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×12], C3⋊C8 [×4], Dic6 [×4], Dic6 [×6], C2×Dic3 [×6], C2×C12 [×6], C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×6], C22×C6, C22×C8, C2×Q16 [×12], C22×Q8, C22×Q8, C2×C3⋊C8 [×6], C3⋊Q16 [×16], C2×Dic6 [×6], C2×Dic6 [×3], C22×Dic3, C22×C12, C22×C12, C6×Q8 [×6], C6×Q8 [×3], C22×Q16, C22×C3⋊C8, C2×C3⋊Q16 [×12], C22×Dic6, Q8×C2×C6, C22×C3⋊Q16

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], Q16 [×4], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C2×Q16 [×6], C22×D4, C3⋊Q16 [×4], C2×C3⋊D4 [×6], S3×C23, C22×Q16, C2×C3⋊Q16 [×6], C22×C3⋊D4, C22×C3⋊Q16

Generators and relations
 G = < a,b,c,d,e | a2=b2=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

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

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

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

(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)

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

Matrix representation ►G ⊆ GL6(𝔽73)

7200000
0720000
0072000
0007200
0000720
0000072
,
7200000
0720000
001000
000100
0000720
0000072
,
0720000
1720000
0007200
0017200
000010
000001
,
1720000
0720000
0017200
0007200
0000032
00005732
,
7200000
0720000
0072000
0007200
00006759
0000606

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,57,0,0,0,0,32,32],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,67,60,0,0,0,0,59,6] >;
 

48 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E4F4G4H4I4J4K4L6A···6G8A···8H12A···12L
order12···234444444444446···68···812···12
size11···1222224444121212122···26···64···4

48 irreducible representations

dim11111222222224
type++++++++++--
imageC1C2C2C2C2S3D4D4D6D6Q16C3⋊D4C3⋊D4C3⋊Q16
kernelC22×C3⋊Q16C22×C3⋊C8C2×C3⋊Q16C22×Dic6Q8×C2×C6C22×Q8C2×C12C22×C6C22×C4C2×Q8C2×C6C2×C4C23C22
# reps111211131168624

In GAP, Magma, Sage, TeX 

C_2^2\times C_3\rtimes Q_{16}
 
% in TeX
 
G:=Group("C2^2xC3:Q16");
 
// GroupNames label
 
G:=SmallGroup(192,1368);
 
// by ID
 
G=gap.SmallGroup(192,1368);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,675,136,1684,235,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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