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## G = C6.67(C4×D4)  order 192 = 26·3

### 7th non-split extension by C6 of C4×D4 acting via C4×D4/C22×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.67(C4×D4)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — C6.67(C4×D4)
 Lower central C3 — C2×C6 — C6.67(C4×D4)
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C6.67(C4×D4)
G = < a,b,c,d | a6=b4=c4=1, d2=a3, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 344 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C3, C4 [×12], C22 [×7], C6 [×7], C2×C4 [×4], C2×C4 [×22], C23, Dic3 [×7], C12 [×5], C2×C6 [×7], C42 [×2], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×6], C2×Dic3 [×9], C2×C12 [×4], C2×C12 [×7], C22×C6, C2.C42 [×4], C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Dic3 [×2], Dic3⋊C4 [×4], C3×C4⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×3], C23.63C23, C6.C42 [×4], C2×C4×Dic3, C2×Dic3⋊C4, C6×C4⋊C4, C6.67(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, S3×C2×C4, C4○D12, D42S3 [×2], S3×Q8, Q83S3, C2×C3⋊D4, C23.63C23, Dic6⋊C4, Dic3.Q8, C4⋊C47S3, C4⋊C4⋊S3, C4×C3⋊D4, C23.23D6, D63Q8, C6.67(C4×D4)

Smallest permutation representation of C6.67(C4×D4)
Regular action on 192 points
Generators in S192
(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 37 13 25)(2 38 14 26)(3 39 15 27)(4 40 16 28)(5 41 17 29)(6 42 18 30)(7 43 19 31)(8 44 20 32)(9 45 21 33)(10 46 22 34)(11 47 23 35)(12 48 24 36)(49 88 61 76)(50 89 62 77)(51 90 63 78)(52 85 64 73)(53 86 65 74)(54 87 66 75)(55 94 67 82)(56 95 68 83)(57 96 69 84)(58 91 70 79)(59 92 71 80)(60 93 72 81)(97 133 109 121)(98 134 110 122)(99 135 111 123)(100 136 112 124)(101 137 113 125)(102 138 114 126)(103 139 115 127)(104 140 116 128)(105 141 117 129)(106 142 118 130)(107 143 119 131)(108 144 120 132)(145 184 157 172)(146 185 158 173)(147 186 159 174)(148 181 160 169)(149 182 161 170)(150 183 162 171)(151 190 163 178)(152 191 164 179)(153 192 165 180)(154 187 166 175)(155 188 167 176)(156 189 168 177)
(1 161 7 167)(2 160 8 166)(3 159 9 165)(4 158 10 164)(5 157 11 163)(6 162 12 168)(13 149 19 155)(14 148 20 154)(15 147 21 153)(16 146 22 152)(17 145 23 151)(18 150 24 156)(25 185 31 191)(26 184 32 190)(27 183 33 189)(28 182 34 188)(29 181 35 187)(30 186 36 192)(37 173 43 179)(38 172 44 178)(39 171 45 177)(40 170 46 176)(41 169 47 175)(42 174 48 180)(49 110 55 116)(50 109 56 115)(51 114 57 120)(52 113 58 119)(53 112 59 118)(54 111 60 117)(61 98 67 104)(62 97 68 103)(63 102 69 108)(64 101 70 107)(65 100 71 106)(66 99 72 105)(73 134 79 140)(74 133 80 139)(75 138 81 144)(76 137 82 143)(77 136 83 142)(78 135 84 141)(85 122 91 128)(86 121 92 127)(87 126 93 132)(88 125 94 131)(89 124 95 130)(90 123 96 129)
(1 52 4 49)(2 53 5 50)(3 54 6 51)(7 58 10 55)(8 59 11 56)(9 60 12 57)(13 64 16 61)(14 65 17 62)(15 66 18 63)(19 70 22 67)(20 71 23 68)(21 72 24 69)(25 76 28 73)(26 77 29 74)(27 78 30 75)(31 82 34 79)(32 83 35 80)(33 84 36 81)(37 88 40 85)(38 89 41 86)(39 90 42 87)(43 94 46 91)(44 95 47 92)(45 96 48 93)(97 154 100 151)(98 155 101 152)(99 156 102 153)(103 148 106 145)(104 149 107 146)(105 150 108 147)(109 166 112 163)(110 167 113 164)(111 168 114 165)(115 160 118 157)(116 161 119 158)(117 162 120 159)(121 178 124 175)(122 179 125 176)(123 180 126 177)(127 172 130 169)(128 173 131 170)(129 174 132 171)(133 190 136 187)(134 191 137 188)(135 192 138 189)(139 184 142 181)(140 185 143 182)(141 186 144 183)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + - - + image C1 C2 C2 C2 C2 C4 S3 Q8 D4 D6 C4○D4 C4×S3 C3⋊D4 C4○D12 D4⋊2S3 S3×Q8 Q8⋊3S3 kernel C6.67(C4×D4) C6.C42 C2×C4×Dic3 C2×Dic3⋊C4 C6×C4⋊C4 Dic3⋊C4 C2×C4⋊C4 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 C22 # reps 1 4 1 1 1 8 1 2 2 3 8 4 4 4 2 1 1

Matrix representation of C6.67(C4×D4) in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 0 0 0 0 0 0 9
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 8 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 3 0 0 0 0 4 0
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,4,0,0,0,0,3,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;

C6.67(C4×D4) in GAP, Magma, Sage, TeX

C_6._{67}(C_4\times D_4)
% in TeX

G:=Group("C6.67(C4xD4)");
// GroupNames label

G:=SmallGroup(192,537);
// by ID

G=gap.SmallGroup(192,537);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=c^4=1,d^2=a^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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