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## G = C8×C3⋊C8order 192 = 26·3

### Direct product of C8 and C3⋊C8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C8×C3⋊C8
G = < a,b,c | a8=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 104 in 74 conjugacy classes, 59 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, C12, C2×C6, C42, C2×C8, C2×C8, C3⋊C8, C24, C2×C12, C2×C12, C4×C8, C4×C8, C2×C3⋊C8, C4×C12, C2×C24, C82, C4×C3⋊C8, C4×C24, C8×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, C3⋊C8, C4×S3, C2×Dic3, C4×C8, S3×C8, C2×C3⋊C8, C4×Dic3, C82, C4×C3⋊C8, C8×Dic3, C8×C3⋊C8

Smallest permutation representation of C8×C3⋊C8
Regular action on 192 points
Generators in S192
(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 25 67)(2 26 68)(3 27 69)(4 28 70)(5 29 71)(6 30 72)(7 31 65)(8 32 66)(9 129 121)(10 130 122)(11 131 123)(12 132 124)(13 133 125)(14 134 126)(15 135 127)(16 136 128)(17 85 95)(18 86 96)(19 87 89)(20 88 90)(21 81 91)(22 82 92)(23 83 93)(24 84 94)(33 47 147)(34 48 148)(35 41 149)(36 42 150)(37 43 151)(38 44 152)(39 45 145)(40 46 146)(49 169 59)(50 170 60)(51 171 61)(52 172 62)(53 173 63)(54 174 64)(55 175 57)(56 176 58)(73 97 111)(74 98 112)(75 99 105)(76 100 106)(77 101 107)(78 102 108)(79 103 109)(80 104 110)(113 191 181)(114 192 182)(115 185 183)(116 186 184)(117 187 177)(118 188 178)(119 189 179)(120 190 180)(137 165 154)(138 166 155)(139 167 156)(140 168 157)(141 161 158)(142 162 159)(143 163 160)(144 164 153)
(1 174 79 116 21 12 145 142)(2 175 80 117 22 13 146 143)(3 176 73 118 23 14 147 144)(4 169 74 119 24 15 148 137)(5 170 75 120 17 16 149 138)(6 171 76 113 18 9 150 139)(7 172 77 114 19 10 151 140)(8 173 78 115 20 11 152 141)(25 54 103 184 81 124 39 159)(26 55 104 177 82 125 40 160)(27 56 97 178 83 126 33 153)(28 49 98 179 84 127 34 154)(29 50 99 180 85 128 35 155)(30 51 100 181 86 121 36 156)(31 52 101 182 87 122 37 157)(32 53 102 183 88 123 38 158)(41 166 71 60 105 190 95 136)(42 167 72 61 106 191 96 129)(43 168 65 62 107 192 89 130)(44 161 66 63 108 185 90 131)(45 162 67 64 109 186 91 132)(46 163 68 57 110 187 92 133)(47 164 69 58 111 188 93 134)(48 165 70 59 112 189 94 135)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4L 6A 6B 6C 8A ··· 8P 8Q ··· 8AV 12A ··· 12L 24A ··· 24P order 1 2 2 2 3 4 ··· 4 6 6 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 1 ··· 1 2 2 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C4 C4 C8 C8 S3 D6 Dic3 C3⋊C8 C4×S3 S3×C8 kernel C8×C3⋊C8 C4×C3⋊C8 C4×C24 C2×C3⋊C8 C2×C24 C3⋊C8 C24 C4×C8 C42 C2×C8 C8 C2×C4 C4 # reps 1 2 1 8 4 32 16 1 1 2 8 4 16

Matrix representation of C8×C3⋊C8 in GL3(𝔽73) generated by

 63 0 0 0 1 0 0 0 1
,
 1 0 0 0 0 72 0 1 72
,
 10 0 0 0 28 72 0 27 45
G:=sub<GL(3,GF(73))| [63,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,72,72],[10,0,0,0,28,27,0,72,45] >;

C8×C3⋊C8 in GAP, Magma, Sage, TeX

C_8\times C_3\rtimes C_8
% in TeX

G:=Group("C8xC3:C8");
// GroupNames label

G:=SmallGroup(192,12);
// by ID

G=gap.SmallGroup(192,12);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,64,100,136,6278]);
// Polycyclic

G:=Group<a,b,c|a^8=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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