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## G = C2×C24⋊C4order 192 = 26·3

### Direct product of C2 and C24⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C24⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — C2×C4×Dic3 — C2×C24⋊C4
 Lower central C3 — C6 — C2×C24⋊C4
 Upper central C1 — C22×C4 — C22×C8

Generators and relations for C2×C24⋊C4
G = < a,b,c | a2=b24=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 248 in 146 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C8⋊C4, C2×C42, C22×C8, C22×C8, C2×C3⋊C8, C4×Dic3, C2×C24, C22×Dic3, C22×C12, C2×C8⋊C4, C24⋊C4, C22×C3⋊C8, C2×C4×Dic3, C22×C24, C2×C24⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, M4(2), C22×C4, C4×S3, C2×Dic3, C22×S3, C8⋊C4, C2×C42, C2×M4(2), C8⋊S3, C4×Dic3, S3×C2×C4, C22×Dic3, C2×C8⋊C4, C24⋊C4, C2×C8⋊S3, C2×C4×Dic3, C2×C24⋊C4

Smallest permutation representation of C2×C24⋊C4
Regular action on 192 points
Generators in S192
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 167)(26 168)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 159)(42 160)(43 161)(44 162)(45 163)(46 164)(47 165)(48 166)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 130)(82 131)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 121)(97 176)(98 177)(99 178)(100 179)(101 180)(102 181)(103 182)(104 183)(105 184)(106 185)(107 186)(108 187)(109 188)(110 189)(111 190)(112 191)(113 192)(114 169)(115 170)(116 171)(117 172)(118 173)(119 174)(120 175)
(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)
(1 143 170 45)(2 124 171 26)(3 129 172 31)(4 134 173 36)(5 139 174 41)(6 144 175 46)(7 125 176 27)(8 130 177 32)(9 135 178 37)(10 140 179 42)(11 121 180 47)(12 126 181 28)(13 131 182 33)(14 136 183 38)(15 141 184 43)(16 122 185 48)(17 127 186 29)(18 132 187 34)(19 137 188 39)(20 142 189 44)(21 123 190 25)(22 128 191 30)(23 133 192 35)(24 138 169 40)(49 94 115 163)(50 75 116 168)(51 80 117 149)(52 85 118 154)(53 90 119 159)(54 95 120 164)(55 76 97 145)(56 81 98 150)(57 86 99 155)(58 91 100 160)(59 96 101 165)(60 77 102 146)(61 82 103 151)(62 87 104 156)(63 92 105 161)(64 73 106 166)(65 78 107 147)(66 83 108 152)(67 88 109 157)(68 93 110 162)(69 74 111 167)(70 79 112 148)(71 84 113 153)(72 89 114 158)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4P 6A ··· 6G 8A ··· 8H 8I ··· 8P 12A ··· 12H 24A ··· 24P order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 2 1 ··· 1 6 ··· 6 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 S3 Dic3 D6 D6 M4(2) C4×S3 C4×S3 C8⋊S3 kernel C2×C24⋊C4 C24⋊C4 C22×C3⋊C8 C2×C4×Dic3 C22×C24 C2×C3⋊C8 C4×Dic3 C2×C24 C22×Dic3 C22×C8 C2×C8 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 4 1 1 1 8 4 8 4 1 4 2 1 8 6 2 16

Matrix representation of C2×C24⋊C4 in GL4(𝔽73) generated by

 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 27 0 0 0 0 72 0 0 0 0 67 70 0 0 3 70
,
 27 0 0 0 0 46 0 0 0 0 2 55 0 0 53 71
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,72,0,0,0,0,67,3,0,0,70,70],[27,0,0,0,0,46,0,0,0,0,2,53,0,0,55,71] >;

C2×C24⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{24}\rtimes C_4
% in TeX

G:=Group("C2xC24:C4");
// GroupNames label

G:=SmallGroup(192,659);
// by ID

G=gap.SmallGroup(192,659);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,100,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^2=b^24=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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