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G = C23⋊C8order 184 = 23·23

The semidirect product of C23 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C23⋊C8, C46.C4, C92.2C2, C4.2D23, C2.Dic23, SmallGroup(184,1)

Series: Derived Chief Lower central Upper central

C1C23 — C23⋊C8
C1C23C46C92 — C23⋊C8
C23 — C23⋊C8
C1C4

Generators and relations for C23⋊C8
 G = < a,b | a23=b8=1, bab-1=a-1 >

23C8

Smallest permutation representation of C23⋊C8
Regular action 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)
(1 162 76 116 32 139 58 93)(2 184 77 138 33 161 59 115)(3 183 78 137 34 160 60 114)(4 182 79 136 35 159 61 113)(5 181 80 135 36 158 62 112)(6 180 81 134 37 157 63 111)(7 179 82 133 38 156 64 110)(8 178 83 132 39 155 65 109)(9 177 84 131 40 154 66 108)(10 176 85 130 41 153 67 107)(11 175 86 129 42 152 68 106)(12 174 87 128 43 151 69 105)(13 173 88 127 44 150 47 104)(14 172 89 126 45 149 48 103)(15 171 90 125 46 148 49 102)(16 170 91 124 24 147 50 101)(17 169 92 123 25 146 51 100)(18 168 70 122 26 145 52 99)(19 167 71 121 27 144 53 98)(20 166 72 120 28 143 54 97)(21 165 73 119 29 142 55 96)(22 164 74 118 30 141 56 95)(23 163 75 117 31 140 57 94)

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

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

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

C23⋊C8 is a maximal subgroup of   C8×D23  C8⋊D23  C92.C4  D4⋊D23  D4.D23  Q8⋊D23  C23⋊Q16
C23⋊C8 is a maximal quotient of   C23⋊C16

52 conjugacy classes

class 1  2 4A4B8A8B8C8D23A···23K46A···46K92A···92V
order1244888823···2346···4692···92
size1111232323232···22···22···2

52 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D23Dic23C23⋊C8
kernelC23⋊C8C92C46C23C4C2C1
# reps1124111122

Matrix representation of C23⋊C8 in GL3(𝔽1289) generated by

100
001
01288408
,
88700
02901252
0984999
G:=sub<GL(3,GF(1289))| [1,0,0,0,0,1288,0,1,408],[887,0,0,0,290,984,0,1252,999] >;

C23⋊C8 in GAP, Magma, Sage, TeX

C_{23}\rtimes C_8
% in TeX

G:=Group("C23:C8");
// GroupNames label

G:=SmallGroup(184,1);
// by ID

G=gap.SmallGroup(184,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-23,8,21,2819]);
// Polycyclic

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

Export

Subgroup lattice of C23⋊C8 in TeX

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