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G = D8×C23order 368 = 24·23

Direct product of C23 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: D8×C23, D4⋊C46, C81C46, C1845C2, C46.14D4, C92.17C22, (D4×C23)⋊4C2, C4.1(C2×C46), C2.3(D4×C23), SmallGroup(368,24)

Series: Derived Chief Lower central Upper central

C1C4 — D8×C23
C1C2C4C92D4×C23 — D8×C23
C1C2C4 — D8×C23
C1C46C92 — D8×C23

Generators and relations for D8×C23
 G = < a,b,c | a23=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

4C2
4C2
2C22
2C22
4C46
4C46
2C2×C46
2C2×C46

Smallest permutation representation of D8×C23
On 184 points
Generators in S184
(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)
(1 80 106 58 181 143 132 27)(2 81 107 59 182 144 133 28)(3 82 108 60 183 145 134 29)(4 83 109 61 184 146 135 30)(5 84 110 62 162 147 136 31)(6 85 111 63 163 148 137 32)(7 86 112 64 164 149 138 33)(8 87 113 65 165 150 116 34)(9 88 114 66 166 151 117 35)(10 89 115 67 167 152 118 36)(11 90 93 68 168 153 119 37)(12 91 94 69 169 154 120 38)(13 92 95 47 170 155 121 39)(14 70 96 48 171 156 122 40)(15 71 97 49 172 157 123 41)(16 72 98 50 173 158 124 42)(17 73 99 51 174 159 125 43)(18 74 100 52 175 160 126 44)(19 75 101 53 176 161 127 45)(20 76 102 54 177 139 128 46)(21 77 103 55 178 140 129 24)(22 78 104 56 179 141 130 25)(23 79 105 57 180 142 131 26)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 24)(22 25)(23 26)(47 170)(48 171)(49 172)(50 173)(51 174)(52 175)(53 176)(54 177)(55 178)(56 179)(57 180)(58 181)(59 182)(60 183)(61 184)(62 162)(63 163)(64 164)(65 165)(66 166)(67 167)(68 168)(69 169)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)(82 134)(83 135)(84 136)(85 137)(86 138)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 161)(102 139)(103 140)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)

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

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

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

161 conjugacy classes

class 1 2A2B2C 4 8A8B23A···23V46A···46V46W···46BN92A···92V184A···184AR
order122248823···2346···4646···4692···92184···184
size11442221···11···14···42···22···2

161 irreducible representations

dim1111112222
type+++++
imageC1C2C2C23C46C46D4D8D4×C23D8×C23
kernelD8×C23C184D4×C23D8C8D4C46C23C2C1
# reps112222244122244

Matrix representation of D8×C23 in GL2(𝔽47) generated by

170
017
,
76
390
,
06
80
G:=sub<GL(2,GF(47))| [17,0,0,17],[7,39,6,0],[0,8,6,0] >;

D8×C23 in GAP, Magma, Sage, TeX

D_8\times C_{23}
% in TeX

G:=Group("D8xC23");
// GroupNames label

G:=SmallGroup(368,24);
// by ID

G=gap.SmallGroup(368,24);
# by ID

G:=PCGroup([5,-2,-2,-23,-2,-2,941,5523,2768,58]);
// Polycyclic

G:=Group<a,b,c|a^23=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D8×C23 in TeX

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