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## G = (C6×Q8)⋊7C4order 192 = 26·3

### 3rd semidirect product of C6×Q8 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C6×Q8)⋊7C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — (C6×Q8)⋊7C4
 Lower central C3 — C2×C6 — (C6×Q8)⋊7C4
 Upper central C1 — C23 — C22×Q8

Generators and relations for (C6×Q8)⋊7C4
G = < a,b,c,d | a6=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, dbd-1=a3b-1, cd=dc >

Subgroups: 392 in 186 conjugacy classes, 91 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×10], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×18], Q8 [×8], C23, Dic3 [×6], C12 [×4], C12 [×4], C2×C6 [×3], C2×C6 [×4], C42 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×10], C2×C12 [×4], C3×Q8 [×8], C22×C6, C2.C42 [×4], C2×C42, C2×C4⋊C4, C22×Q8, C4×Dic3 [×2], C4⋊Dic3 [×2], C22×Dic3 [×4], C22×C12, C22×C12 [×2], C6×Q8 [×4], C6×Q8 [×4], C23.67C23, C6.C42 [×4], C2×C4×Dic3, C2×C4⋊Dic3, Q8×C2×C6, (C6×Q8)⋊7C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], Q8 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C6.D4 [×4], S3×Q8 [×2], Q83S3 [×2], C22×Dic3, C2×C3⋊D4 [×2], C23.67C23, Dic3⋊Q8, Q8×Dic3 [×2], D63Q8 [×2], C12.23D4, C2×C6.D4, (C6×Q8)⋊7C4

Smallest permutation representation of (C6×Q8)⋊7C4
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 40 17 35)(2 41 18 36)(3 42 13 31)(4 37 14 32)(5 38 15 33)(6 39 16 34)(7 166 190 170)(8 167 191 171)(9 168 192 172)(10 163 187 173)(11 164 188 174)(12 165 189 169)(19 46 30 51)(20 47 25 52)(21 48 26 53)(22 43 27 54)(23 44 28 49)(24 45 29 50)(55 87 66 82)(56 88 61 83)(57 89 62 84)(58 90 63 79)(59 85 64 80)(60 86 65 81)(67 98 78 94)(68 99 73 95)(69 100 74 96)(70 101 75 91)(71 102 76 92)(72 97 77 93)(103 137 113 127)(104 138 114 128)(105 133 109 129)(106 134 110 130)(107 135 111 131)(108 136 112 132)(115 149 125 139)(116 150 126 140)(117 145 121 141)(118 146 122 142)(119 147 123 143)(120 148 124 144)(151 175 161 185)(152 176 162 186)(153 177 157 181)(154 178 158 182)(155 179 159 183)(156 180 160 184)
(1 76 17 71)(2 77 18 72)(3 78 13 67)(4 73 14 68)(5 74 15 69)(6 75 16 70)(7 127 190 137)(8 128 191 138)(9 129 192 133)(10 130 187 134)(11 131 188 135)(12 132 189 136)(19 58 30 63)(20 59 25 64)(21 60 26 65)(22 55 27 66)(23 56 28 61)(24 57 29 62)(31 94 42 98)(32 95 37 99)(33 96 38 100)(34 91 39 101)(35 92 40 102)(36 93 41 97)(43 82 54 87)(44 83 49 88)(45 84 50 89)(46 79 51 90)(47 80 52 85)(48 81 53 86)(103 166 113 170)(104 167 114 171)(105 168 109 172)(106 163 110 173)(107 164 111 174)(108 165 112 169)(115 154 125 158)(116 155 126 159)(117 156 121 160)(118 151 122 161)(119 152 123 162)(120 153 124 157)(139 178 149 182)(140 179 150 183)(141 180 145 184)(142 175 146 185)(143 176 147 186)(144 177 148 181)
(1 122 25 110)(2 121 26 109)(3 126 27 114)(4 125 28 113)(5 124 29 112)(6 123 30 111)(7 102 182 85)(8 101 183 90)(9 100 184 89)(10 99 185 88)(11 98 186 87)(12 97 181 86)(13 116 22 104)(14 115 23 103)(15 120 24 108)(16 119 19 107)(17 118 20 106)(18 117 21 105)(31 143 43 131)(32 142 44 130)(33 141 45 129)(34 140 46 128)(35 139 47 127)(36 144 48 132)(37 146 49 134)(38 145 50 133)(39 150 51 138)(40 149 52 137)(41 148 53 136)(42 147 54 135)(55 167 67 155)(56 166 68 154)(57 165 69 153)(58 164 70 152)(59 163 71 151)(60 168 72 156)(61 170 73 158)(62 169 74 157)(63 174 75 162)(64 173 76 161)(65 172 77 160)(66 171 78 159)(79 191 91 179)(80 190 92 178)(81 189 93 177)(82 188 94 176)(83 187 95 175)(84 192 96 180)

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

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

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

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 4 4 type + + + + + + - + + - - + image C1 C2 C2 C2 C2 C4 S3 Q8 D4 D6 Dic3 C4○D4 C3⋊D4 S3×Q8 Q8⋊3S3 kernel (C6×Q8)⋊7C4 C6.C42 C2×C4×Dic3 C2×C4⋊Dic3 Q8×C2×C6 C6×Q8 C22×Q8 C2×Dic3 C2×C12 C22×C4 C2×Q8 C2×C6 C2×C4 C22 C22 # reps 1 4 1 1 1 8 1 4 4 3 4 4 8 2 2

Matrix representation of (C6×Q8)⋊7C4 in GL5(𝔽13)

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

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

(C6×Q8)⋊7C4 in GAP, Magma, Sage, TeX

(C_6\times Q_8)\rtimes_7C_4
% in TeX

G:=Group("(C6xQ8):7C4");
// GroupNames label

G:=SmallGroup(192,788);
// by ID

G=gap.SmallGroup(192,788);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,387,184,6278]);
// Polycyclic

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

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