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G = C8⋊D23order 368 = 24·23

3rd semidirect product of C8 and D23 acting via D23/C23=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D23, D46.C4, C1844C2, C4.13D46, Dic23.C4, C231M4(2), C92.13C22, C23⋊C84C2, C46.2(C2×C4), C2.3(C4×D23), (C4×D23).2C2, SmallGroup(368,4)

Series: Derived Chief Lower central Upper central

C1C46 — C8⋊D23
C1C23C46C92C4×D23 — C8⋊D23
C23C46 — C8⋊D23
C1C4C8

Generators and relations for C8⋊D23
 G = < a,b,c | a8=b23=c2=1, ab=ba, cac=a5, cbc=b-1 >

46C2
23C22
23C4
2D23
23C2×C4
23C8
23M4(2)

Smallest permutation representation of C8⋊D23
On 184 points
Generators in S184
(1 181 83 127 39 150 68 111)(2 182 84 128 40 151 69 112)(3 183 85 129 41 152 47 113)(4 184 86 130 42 153 48 114)(5 162 87 131 43 154 49 115)(6 163 88 132 44 155 50 93)(7 164 89 133 45 156 51 94)(8 165 90 134 46 157 52 95)(9 166 91 135 24 158 53 96)(10 167 92 136 25 159 54 97)(11 168 70 137 26 160 55 98)(12 169 71 138 27 161 56 99)(13 170 72 116 28 139 57 100)(14 171 73 117 29 140 58 101)(15 172 74 118 30 141 59 102)(16 173 75 119 31 142 60 103)(17 174 76 120 32 143 61 104)(18 175 77 121 33 144 62 105)(19 176 78 122 34 145 63 106)(20 177 79 123 35 146 64 107)(21 178 80 124 36 147 65 108)(22 179 81 125 37 148 66 109)(23 180 82 126 38 149 67 110)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 30)(25 29)(26 28)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(66 69)(67 68)(70 72)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(93 121)(94 120)(95 119)(96 118)(97 117)(98 116)(99 138)(100 137)(101 136)(102 135)(103 134)(104 133)(105 132)(106 131)(107 130)(108 129)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(139 168)(140 167)(141 166)(142 165)(143 164)(144 163)(145 162)(146 184)(147 183)(148 182)(149 181)(150 180)(151 179)(152 178)(153 177)(154 176)(155 175)(156 174)(157 173)(158 172)(159 171)(160 170)(161 169)

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

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

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

98 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D23A···23K46A···46K92A···92V184A···184AR
order122444888823···2346···4692···92184···184
size114611462246462···22···22···22···2

98 irreducible representations

dim11111122222
type++++++
imageC1C2C2C2C4C4M4(2)D23D46C4×D23C8⋊D23
kernelC8⋊D23C23⋊C8C184C4×D23Dic23D46C23C8C4C2C1
# reps111122211112244

Matrix representation of C8⋊D23 in GL2(𝔽1289) generated by

235614
6751054
,
5341
12880
,
01
10
G:=sub<GL(2,GF(1289))| [235,675,614,1054],[534,1288,1,0],[0,1,1,0] >;

C8⋊D23 in GAP, Magma, Sage, TeX

C_8\rtimes D_{23}
% in TeX

G:=Group("C8:D23");
// GroupNames label

G:=SmallGroup(368,4);
// by ID

G=gap.SmallGroup(368,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-23,101,26,42,8804]);
// Polycyclic

G:=Group<a,b,c|a^8=b^23=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D23 in TeX

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