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G = C2.Dic24order 192 = 26·3

1st central extension by C2 of Dic24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C2.Dic24
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C24⋊1C4 — C2.Dic24
 Lower central C3 — C6 — C12 — C24 — C2.Dic24
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2.Dic24
G = < a,b,c | a2=b48=1, c2=ab24, ab=ba, ac=ca, cbc-1=ab-1 >

Subgroups: 200 in 58 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C16, C4⋊C4, C2×C8, Q16, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2.D8, C2×C16, C2×Q16, C48, Dic12, Dic12, C4⋊Dic3, C2×C24, C2×Dic6, C2.Q32, C241C4, C2×C48, C2×Dic12, C2.Dic24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, SD32, Q32, C24⋊C2, D24, D6⋊C4, C2.Q32, C48⋊C2, Dic24, C2.D24, C2.Dic24

Smallest permutation representation of C2.Dic24
Regular action on 192 points
Generators in S192
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 181)(26 182)(27 183)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 145)(38 146)(39 147)(40 148)(41 149)(42 150)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 143)(50 144)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 105)(60 106)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)
(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 83 181 105)(2 128 182 58)(3 81 183 103)(4 126 184 56)(5 79 185 101)(6 124 186 54)(7 77 187 99)(8 122 188 52)(9 75 189 97)(10 120 190 50)(11 73 191 143)(12 118 192 96)(13 71 145 141)(14 116 146 94)(15 69 147 139)(16 114 148 92)(17 67 149 137)(18 112 150 90)(19 65 151 135)(20 110 152 88)(21 63 153 133)(22 108 154 86)(23 61 155 131)(24 106 156 84)(25 59 157 129)(26 104 158 82)(27 57 159 127)(28 102 160 80)(29 55 161 125)(30 100 162 78)(31 53 163 123)(32 98 164 76)(33 51 165 121)(34 144 166 74)(35 49 167 119)(36 142 168 72)(37 95 169 117)(38 140 170 70)(39 93 171 115)(40 138 172 68)(41 91 173 113)(42 136 174 66)(43 89 175 111)(44 134 176 64)(45 87 177 109)(46 132 178 62)(47 85 179 107)(48 130 180 60)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 16A ··· 16H 24A ··· 24H 48A ··· 48P order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 2 2 2 24 24 24 24 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + - + - image C1 C2 C2 C2 C4 S3 D4 D4 D6 SD16 D8 C4×S3 C3⋊D4 D12 SD32 Q32 C24⋊C2 D24 C48⋊C2 Dic24 kernel C2.Dic24 C24⋊1C4 C2×C48 C2×Dic12 Dic12 C2×C16 C24 C2×C12 C2×C8 C12 C2×C6 C8 C8 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 4 4 4 4 8 8

Matrix representation of C2.Dic24 in GL3(𝔽97) generated by

 96 0 0 0 96 0 0 0 96
,
 75 0 0 0 2 54 0 43 45
,
 75 0 0 0 36 93 0 57 61
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[75,0,0,0,2,43,0,54,45],[75,0,0,0,36,57,0,93,61] >;

C2.Dic24 in GAP, Magma, Sage, TeX

C_2.{\rm Dic}_{24}
% in TeX

G:=Group("C2.Dic24");
// GroupNames label

G:=SmallGroup(192,62);
// by ID

G=gap.SmallGroup(192,62);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,280,85,204,422,268,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^2=b^48=1,c^2=a*b^24,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations

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