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## G = (C2×C24)⋊5C4order 192 = 26·3

### 1st semidirect product of C2×C24 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — (C2×C24)⋊5C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C2×C4×Dic3 — (C2×C24)⋊5C4
 Lower central C3 — C6 — (C2×C24)⋊5C4
 Upper central C1 — C22×C4 — C22×C8

Generators and relations for (C2×C24)⋊5C4
G = < a,b,c | a2=b24=c4=1, ab=ba, ac=ca, cbc-1=ab17 >

Subgroups: 232 in 118 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C2×C42, C22×C8, C22×C8, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C2×C24, C2×C24, C22×Dic3, C22×C12, C22.7C42, C22×C3⋊C8, C2×C4×Dic3, C22×C24, (C2×C24)⋊5C4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, S3×C8, C8⋊S3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C22.7C42, C8×Dic3, Dic3⋊C8, C24⋊C4, D6⋊C8, C6.C42, (C2×C24)⋊5C4

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

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

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

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

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 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - - + - + image C1 C2 C2 C2 C4 C4 C4 C8 S3 D4 Q8 Dic3 D6 M4(2) Dic6 C4×S3 D12 C3⋊D4 C4×S3 S3×C8 C8⋊S3 kernel (C2×C24)⋊5C4 C22×C3⋊C8 C2×C4×Dic3 C22×C24 C2×C3⋊C8 C2×C24 C22×Dic3 C2×Dic3 C22×C8 C2×C12 C2×C12 C2×C8 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 1 1 1 4 4 4 16 1 3 1 2 1 4 2 2 2 4 2 8 8

Matrix representation of (C2×C24)⋊5C4 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 22 0 0 0 0 0 22 22 0 0 0 51 0 0 0 0 0 0 3 70 0 0 0 3 6
,
 46 0 0 0 0 0 60 71 0 0 0 11 13 0 0 0 0 0 14 19 0 0 0 5 59

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[22,0,0,0,0,0,22,51,0,0,0,22,0,0,0,0,0,0,3,3,0,0,0,70,6],[46,0,0,0,0,0,60,11,0,0,0,71,13,0,0,0,0,0,14,5,0,0,0,19,59] >;

(C2×C24)⋊5C4 in GAP, Magma, Sage, TeX

(C_2\times C_{24})\rtimes_5C_4
% in TeX

G:=Group("(C2xC24):5C4");
// GroupNames label

G:=SmallGroup(192,109);
// by ID

G=gap.SmallGroup(192,109);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,136,6278]);
// 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=a*b^17>;
// generators/relations

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