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## G = C4.(D6⋊C4)  order 192 = 26·3

### 9th non-split extension by C4 of D6⋊C4 acting via D6⋊C4/C2×Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C4.(D6⋊C4)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C4.(D6⋊C4)
 Lower central C3 — C2×C6 — C4.(D6⋊C4)
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C4.(D6⋊C4)
G = < a,b,c,d | a4=b6=d4=1, c2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 440 in 186 conjugacy classes, 83 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C4×Dic3, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C22×C12, C23.67C23, C6.C42, C2×C4×Dic3, C6×C4⋊C4, C22×Dic6, C4.(D6⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, D6⋊C4, S3×C2×C4, C2×D12, D42S3, S3×Q8, C2×C3⋊D4, C23.67C23, Dic6⋊C4, C4.D12, C2×D6⋊C4, C23.12D6, Dic3⋊Q8, C4.(D6⋊C4)

Smallest permutation representation of C4.(D6⋊C4)
Regular action on 192 points
Generators in S192
```(1 55 7 63)(2 56 8 64)(3 57 9 65)(4 58 10 66)(5 59 11 61)(6 60 12 62)(13 146 190 142)(14 147 191 143)(15 148 192 144)(16 149 187 139)(17 150 188 140)(18 145 189 141)(19 78 30 67)(20 73 25 68)(21 74 26 69)(22 75 27 70)(23 76 28 71)(24 77 29 72)(31 87 39 79)(32 88 40 80)(33 89 41 81)(34 90 42 82)(35 85 37 83)(36 86 38 84)(43 101 54 91)(44 102 49 92)(45 97 50 93)(46 98 51 94)(47 99 52 95)(48 100 53 96)(103 159 111 151)(104 160 112 152)(105 161 113 153)(106 162 114 154)(107 157 109 155)(108 158 110 156)(115 173 125 163)(116 174 126 164)(117 169 121 165)(118 170 122 166)(119 171 123 167)(120 172 124 168)(127 183 135 175)(128 184 136 176)(129 185 137 177)(130 186 138 178)(131 181 133 179)(132 182 134 180)
(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)
(1 145 7 141)(2 150 8 140)(3 149 9 139)(4 148 10 144)(5 147 11 143)(6 146 12 142)(13 62 190 60)(14 61 191 59)(15 66 192 58)(16 65 187 57)(17 64 188 56)(18 63 189 55)(19 132 30 134)(20 131 25 133)(21 130 26 138)(22 129 27 137)(23 128 28 136)(24 127 29 135)(31 107 39 109)(32 106 40 114)(33 105 41 113)(34 104 42 112)(35 103 37 111)(36 108 38 110)(43 118 54 122)(44 117 49 121)(45 116 50 126)(46 115 51 125)(47 120 52 124)(48 119 53 123)(67 182 78 180)(68 181 73 179)(69 186 74 178)(70 185 75 177)(71 184 76 176)(72 183 77 175)(79 157 87 155)(80 162 88 154)(81 161 89 153)(82 160 90 152)(83 159 85 151)(84 158 86 156)(91 170 101 166)(92 169 102 165)(93 174 97 164)(94 173 98 163)(95 172 99 168)(96 171 100 167)
(1 49 25 39)(2 50 26 40)(3 51 27 41)(4 52 28 42)(5 53 29 37)(6 54 30 38)(7 44 20 31)(8 45 21 32)(9 46 22 33)(10 47 23 34)(11 48 24 35)(12 43 19 36)(13 163 182 153)(14 164 183 154)(15 165 184 155)(16 166 185 156)(17 167 186 151)(18 168 181 152)(55 102 68 87)(56 97 69 88)(57 98 70 89)(58 99 71 90)(59 100 72 85)(60 101 67 86)(61 96 77 83)(62 91 78 84)(63 92 73 79)(64 93 74 80)(65 94 75 81)(66 95 76 82)(103 140 119 130)(104 141 120 131)(105 142 115 132)(106 143 116 127)(107 144 117 128)(108 139 118 129)(109 148 121 136)(110 149 122 137)(111 150 123 138)(112 145 124 133)(113 146 125 134)(114 147 126 135)(157 192 169 176)(158 187 170 177)(159 188 171 178)(160 189 172 179)(161 190 173 180)(162 191 174 175)```

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

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

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

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 type + + + + + + - + + + - - image C1 C2 C2 C2 C2 C4 S3 Q8 D4 D6 C4○D4 C4×S3 D12 C3⋊D4 D4⋊2S3 S3×Q8 kernel C4.(D6⋊C4) C6.C42 C2×C4×Dic3 C6×C4⋊C4 C22×Dic6 C2×Dic6 C2×C4⋊C4 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 4 1 1 1 8 1 4 4 3 4 4 4 4 2 2

Matrix representation of C4.(D6⋊C4) in GL6(𝔽13)

 3 9 0 0 0 0 9 10 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 10 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 5 1 0 0 0 0 2 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8

`G:=sub<GL(6,GF(13))| [3,9,0,0,0,0,9,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,12,10,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,2,0,0,0,0,1,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;`

C4.(D6⋊C4) in GAP, Magma, Sage, TeX

`C_4.(D_6\rtimes C_4)`
`% in TeX`

`G:=Group("C4.(D6:C4)");`
`// GroupNames label`

`G:=SmallGroup(192,532);`
`// by ID`

`G=gap.SmallGroup(192,532);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,310,6278]);`
`// Polycyclic`

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

׿
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