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## G = C8×C24order 192 = 26·3

### Abelian group of type [8,24]

Aliases: C8×C24, SmallGroup(192,127)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8×C24
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C12 — C4×C24 — C8×C24
 Lower central C1 — C8×C24
 Upper central C1 — C8×C24

Generators and relations for C8×C24
G = < a,b | a8=b24=1, ab=ba >

Subgroups: 74, all normal (8 characteristic)
C1, C2 [×3], C3, C4 [×6], C22, C6 [×3], C8 [×12], C2×C4 [×3], C12 [×6], C2×C6, C42, C2×C8 [×6], C24 [×12], C2×C12 [×3], C4×C8 [×3], C4×C12, C2×C24 [×6], C82, C4×C24 [×3], C8×C24
Quotients: C1, C2 [×3], C3, C4 [×6], C22, C6 [×3], C8 [×12], C2×C4 [×3], C12 [×6], C2×C6, C42, C2×C8 [×6], C24 [×12], C2×C12 [×3], C4×C8 [×3], C4×C12, C2×C24 [×6], C82, C4×C24 [×3], C8×C24

Smallest permutation representation of C8×C24
Regular action on 192 points
Generators in S192
(1 94 49 31 179 139 149 104)(2 95 50 32 180 140 150 105)(3 96 51 33 181 141 151 106)(4 73 52 34 182 142 152 107)(5 74 53 35 183 143 153 108)(6 75 54 36 184 144 154 109)(7 76 55 37 185 121 155 110)(8 77 56 38 186 122 156 111)(9 78 57 39 187 123 157 112)(10 79 58 40 188 124 158 113)(11 80 59 41 189 125 159 114)(12 81 60 42 190 126 160 115)(13 82 61 43 191 127 161 116)(14 83 62 44 192 128 162 117)(15 84 63 45 169 129 163 118)(16 85 64 46 170 130 164 119)(17 86 65 47 171 131 165 120)(18 87 66 48 172 132 166 97)(19 88 67 25 173 133 167 98)(20 89 68 26 174 134 168 99)(21 90 69 27 175 135 145 100)(22 91 70 28 176 136 146 101)(23 92 71 29 177 137 147 102)(24 93 72 30 178 138 148 103)
(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)

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

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

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

192 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4L 6A ··· 6F 8A ··· 8AV 12A ··· 12X 24A ··· 24CR order 1 2 2 2 3 3 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

192 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C8 C12 C24 kernel C8×C24 C4×C24 C82 C2×C24 C4×C8 C24 C2×C8 C8 # reps 1 3 2 12 6 48 24 96

Matrix representation of C8×C24 in GL2(𝔽73) generated by

 51 0 0 63
,
 70 0 0 21
G:=sub<GL(2,GF(73))| [51,0,0,63],[70,0,0,21] >;

C8×C24 in GAP, Magma, Sage, TeX

C_8\times C_{24}
% in TeX

G:=Group("C8xC24");
// GroupNames label

G:=SmallGroup(192,127);
// by ID

G=gap.SmallGroup(192,127);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,176,136,172]);
// Polycyclic

G:=Group<a,b|a^8=b^24=1,a*b=b*a>;
// generators/relations

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