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## G = C6×Q8⋊C4order 192 = 26·3

### Direct product of C6 and Q8⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C6×Q8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×Q8⋊C4 — C6×Q8⋊C4
 Lower central C1 — C2 — C4 — C6×Q8⋊C4
 Upper central C1 — C22×C6 — C22×C12 — C6×Q8⋊C4

Generators and relations for C6×Q8⋊C4
G = < a,b,c,d | a6=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 242 in 162 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×C6, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C22×C12, C22×C12, C6×Q8, C6×Q8, C2×Q8⋊C4, C3×Q8⋊C4, C6×C4⋊C4, C22×C24, Q8×C2×C6, C6×Q8⋊C4
Quotients:

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

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A 4B 4C 4D 4E ··· 4L 6A ··· 6N 8A ··· 8H 12A ··· 12H 12I ··· 12X 24A ··· 24P order 1 2 ··· 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 ··· 1 1 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + - image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 D4 D4 SD16 Q16 C3×D4 C3×D4 C3×SD16 C3×Q16 kernel C6×Q8⋊C4 C3×Q8⋊C4 C6×C4⋊C4 C22×C24 Q8×C2×C6 C2×Q8⋊C4 C6×Q8 Q8⋊C4 C2×C4⋊C4 C22×C8 C22×Q8 C2×Q8 C2×C12 C22×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 2 8 8 2 2 2 16 3 1 4 4 6 2 8 8

Matrix representation of C6×Q8⋊C4 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 65 0 0 0 0 0 0 65 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 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 72 71 0 0 0 0 0 1 0 0 0 0 0 0 72 2 0 0 0 0 0 1 0 0 0 0 0 0 20 16 0 0 0 0 16 53
,
 62 42 0 0 0 0 11 11 0 0 0 0 0 0 12 55 0 0 0 0 12 61 0 0 0 0 0 0 8 64 0 0 0 0 64 65

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,65,0,0,0,0,0,0,65,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,72,0,0,0,0,0,2,1,0,0,0,0,0,0,20,16,0,0,0,0,16,53],[62,11,0,0,0,0,42,11,0,0,0,0,0,0,12,12,0,0,0,0,55,61,0,0,0,0,0,0,8,64,0,0,0,0,64,65] >;

C6×Q8⋊C4 in GAP, Magma, Sage, TeX

C_6\times Q_8\rtimes C_4
% in TeX

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

G:=SmallGroup(192,848);
// by ID

G=gap.SmallGroup(192,848);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,4204,2111,172]);
// 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,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations

׿
×
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