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## G = (C2×C4).44D12order 192 = 26·3

### 37th non-split extension by C2×C4 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C4).44D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — (C2×C4).44D12
 Lower central C3 — C22×C6 — (C2×C4).44D12
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C4).44D12
G = < a,b,c,d | a2=b12=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 360 in 150 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×11], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×4], C2×C4 [×21], C23, Dic3 [×6], C12 [×5], C2×C6 [×3], C2×C6 [×4], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×4], C2×C12 [×7], C22×C6, C2.C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×3], C23.81C23, C6.C42, C6.C42 [×2], C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C6×C4⋊C4, (C2×C4).44D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], Q8 [×4], C23, D6 [×3], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, C2×D12, C4○D12, D42S3 [×2], S3×Q8 [×2], C2×C3⋊D4, C23.81C23, Dic3.Q8 [×2], C4.D12 [×2], C127D4, C23.23D6, Dic3⋊Q8, (C2×C4).44D12

Smallest permutation representation of (C2×C4).44D12
Regular action on 192 points
Generators in S192
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 121)(13 137)(14 138)(15 139)(16 140)(17 141)(18 142)(19 143)(20 144)(21 133)(22 134)(23 135)(24 136)(25 165)(26 166)(27 167)(28 168)(29 157)(30 158)(31 159)(32 160)(33 161)(34 162)(35 163)(36 164)(37 178)(38 179)(39 180)(40 169)(41 170)(42 171)(43 172)(44 173)(45 174)(46 175)(47 176)(48 177)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 97)(57 98)(58 99)(59 100)(60 101)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 73)(72 74)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(109 190)(110 191)(111 192)(112 181)(113 182)(114 183)(115 184)(116 185)(117 186)(118 187)(119 188)(120 189)
(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)
(1 109 57 140)(2 189 58 15)(3 119 59 138)(4 187 60 13)(5 117 49 136)(6 185 50 23)(7 115 51 134)(8 183 52 21)(9 113 53 144)(10 181 54 19)(11 111 55 142)(12 191 56 17)(14 124 188 100)(16 122 190 98)(18 132 192 108)(20 130 182 106)(22 128 184 104)(24 126 186 102)(25 79 148 170)(26 64 149 40)(27 77 150 180)(28 62 151 38)(29 75 152 178)(30 72 153 48)(31 73 154 176)(32 70 155 46)(33 83 156 174)(34 68 145 44)(35 81 146 172)(36 66 147 42)(37 157 61 86)(39 167 63 96)(41 165 65 94)(43 163 67 92)(45 161 69 90)(47 159 71 88)(74 87 177 158)(76 85 179 168)(78 95 169 166)(80 93 171 164)(82 91 173 162)(84 89 175 160)(97 141 121 110)(99 139 123 120)(101 137 125 118)(103 135 127 116)(105 133 129 114)(107 143 131 112)
(1 172 122 43)(2 171 123 42)(3 170 124 41)(4 169 125 40)(5 180 126 39)(6 179 127 38)(7 178 128 37)(8 177 129 48)(9 176 130 47)(10 175 131 46)(11 174 132 45)(12 173 121 44)(13 166 137 26)(14 165 138 25)(15 164 139 36)(16 163 140 35)(17 162 141 34)(18 161 142 33)(19 160 143 32)(20 159 144 31)(21 158 133 30)(22 157 134 29)(23 168 135 28)(24 167 136 27)(49 77 102 63)(50 76 103 62)(51 75 104 61)(52 74 105 72)(53 73 106 71)(54 84 107 70)(55 83 108 69)(56 82 97 68)(57 81 98 67)(58 80 99 66)(59 79 100 65)(60 78 101 64)(85 116 151 185)(86 115 152 184)(87 114 153 183)(88 113 154 182)(89 112 155 181)(90 111 156 192)(91 110 145 191)(92 109 146 190)(93 120 147 189)(94 119 148 188)(95 118 149 187)(96 117 150 186)

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

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

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

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 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + + + - - image C1 C2 C2 C2 C2 S3 Q8 D4 D6 C4○D4 D12 C3⋊D4 C4○D12 D4⋊2S3 S3×Q8 kernel (C2×C4).44D12 C6.C42 C2×Dic3⋊C4 C2×C4⋊Dic3 C6×C4⋊C4 C2×C4⋊C4 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 3 2 1 1 1 4 4 3 6 4 4 4 2 2

Matrix representation of (C2×C4).44D12 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 3 0 0 0 0 10 6
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 8 5
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 3 7 0 0 0 0 10 10

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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,0,0,5],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,10,0,0,0,0,7,10] >;

(C2×C4).44D12 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{44}D_{12}
% in TeX

G:=Group("(C2xC4).44D12");
// GroupNames label

G:=SmallGroup(192,540);
// by ID

G=gap.SmallGroup(192,540);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,387,184,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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