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## G = C2×C24⋊1C4order 192 = 26·3

### Direct product of C2 and C24⋊1C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C24⋊1C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — C2×C24⋊1C4
 Lower central C3 — C6 — C12 — C2×C24⋊1C4
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C2×C241C4
G = < a,b,c | a2=b24=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 312 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.D8, C2×C4⋊C4, C22×C8, C4⋊Dic3, C4⋊Dic3, C2×C24, C22×Dic3, C22×C12, C2×C2.D8, C241C4, C2×C4⋊Dic3, C22×C24, C2×C241C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, D24, Dic12, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C2×C2.D8, C241C4, C2×D24, C2×Dic12, C2×C4⋊Dic3, C2×C241C4

Smallest permutation representation of C2×C241C4
Regular action on 192 points
Generators in S192
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 190)(50 191)(51 192)(52 169)(53 170)(54 171)(55 172)(56 173)(57 174)(58 175)(59 176)(60 177)(61 178)(62 179)(63 180)(64 181)(65 182)(66 183)(67 184)(68 185)(69 186)(70 187)(71 188)(72 189)(73 157)(74 158)(75 159)(76 160)(77 161)(78 162)(79 163)(80 164)(81 165)(82 166)(83 167)(84 168)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)
(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)
(1 111 60 88)(2 110 61 87)(3 109 62 86)(4 108 63 85)(5 107 64 84)(6 106 65 83)(7 105 66 82)(8 104 67 81)(9 103 68 80)(10 102 69 79)(11 101 70 78)(12 100 71 77)(13 99 72 76)(14 98 49 75)(15 97 50 74)(16 120 51 73)(17 119 52 96)(18 118 53 95)(19 117 54 94)(20 116 55 93)(21 115 56 92)(22 114 57 91)(23 113 58 90)(24 112 59 89)(25 183 166 127)(26 182 167 126)(27 181 168 125)(28 180 145 124)(29 179 146 123)(30 178 147 122)(31 177 148 121)(32 176 149 144)(33 175 150 143)(34 174 151 142)(35 173 152 141)(36 172 153 140)(37 171 154 139)(38 170 155 138)(39 169 156 137)(40 192 157 136)(41 191 158 135)(42 190 159 134)(43 189 160 133)(44 188 161 132)(45 187 162 131)(46 186 163 130)(47 185 164 129)(48 184 165 128)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + + + - - + + + - image C1 C2 C2 C2 C4 S3 D4 Q8 D4 Dic3 D6 D6 D8 Q16 Dic6 D12 D12 D24 Dic12 kernel C2×C24⋊1C4 C24⋊1C4 C2×C4⋊Dic3 C22×C24 C2×C24 C22×C8 C2×C12 C2×C12 C22×C6 C2×C8 C2×C8 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 2 1 8 1 1 2 1 4 2 1 4 4 4 2 2 8 8

Matrix representation of C2×C241C4 in GL4(𝔽73) generated by

 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 55 23 0 0 50 5
,
 27 0 0 0 0 1 0 0 0 0 28 3 0 0 31 45
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,55,50,0,0,23,5],[27,0,0,0,0,1,0,0,0,0,28,31,0,0,3,45] >;

C2×C241C4 in GAP, Magma, Sage, TeX

C_2\times C_{24}\rtimes_1C_4
% in TeX

G:=Group("C2xC24:1C4");
// GroupNames label

G:=SmallGroup(192,664);
// by ID

G=gap.SmallGroup(192,664);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,268,1684,102,6278]);
// 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^-1>;
// generators/relations

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