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G = Q8×C23order 184 = 23·23

Direct product of C23 and Q8

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

Aliases: Q8×C23, C4.C46, C92.3C2, C46.7C22, C2.2(C2×C46), SmallGroup(184,10)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C23
C1C2C46C92 — Q8×C23
C1C2 — Q8×C23
C1C46 — Q8×C23

Generators and relations for Q8×C23
 G = < a,b,c | a23=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C23
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 86 154 42)(2 87 155 43)(3 88 156 44)(4 89 157 45)(5 90 158 46)(6 91 159 24)(7 92 160 25)(8 70 161 26)(9 71 139 27)(10 72 140 28)(11 73 141 29)(12 74 142 30)(13 75 143 31)(14 76 144 32)(15 77 145 33)(16 78 146 34)(17 79 147 35)(18 80 148 36)(19 81 149 37)(20 82 150 38)(21 83 151 39)(22 84 152 40)(23 85 153 41)(47 113 126 179)(48 114 127 180)(49 115 128 181)(50 93 129 182)(51 94 130 183)(52 95 131 184)(53 96 132 162)(54 97 133 163)(55 98 134 164)(56 99 135 165)(57 100 136 166)(58 101 137 167)(59 102 138 168)(60 103 116 169)(61 104 117 170)(62 105 118 171)(63 106 119 172)(64 107 120 173)(65 108 121 174)(66 109 122 175)(67 110 123 176)(68 111 124 177)(69 112 125 178)
(1 57 154 136)(2 58 155 137)(3 59 156 138)(4 60 157 116)(5 61 158 117)(6 62 159 118)(7 63 160 119)(8 64 161 120)(9 65 139 121)(10 66 140 122)(11 67 141 123)(12 68 142 124)(13 69 143 125)(14 47 144 126)(15 48 145 127)(16 49 146 128)(17 50 147 129)(18 51 148 130)(19 52 149 131)(20 53 150 132)(21 54 151 133)(22 55 152 134)(23 56 153 135)(24 105 91 171)(25 106 92 172)(26 107 70 173)(27 108 71 174)(28 109 72 175)(29 110 73 176)(30 111 74 177)(31 112 75 178)(32 113 76 179)(33 114 77 180)(34 115 78 181)(35 93 79 182)(36 94 80 183)(37 95 81 184)(38 96 82 162)(39 97 83 163)(40 98 84 164)(41 99 85 165)(42 100 86 166)(43 101 87 167)(44 102 88 168)(45 103 89 169)(46 104 90 170)

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

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

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

Q8×C23 is a maximal subgroup of   Q8⋊D23  C23⋊Q16  D92⋊C2

115 conjugacy classes

class 1  2 4A4B4C23A···23V46A···46V92A···92BN
order1244423···2346···4692···92
size112221···11···12···2

115 irreducible representations

dim111122
type++-
imageC1C2C23C46Q8Q8×C23
kernelQ8×C23C92Q8C4C23C1
# reps132266122

Matrix representation of Q8×C23 in GL2(𝔽47) generated by

40
04
,
4416
173
,
024
450
G:=sub<GL(2,GF(47))| [4,0,0,4],[44,17,16,3],[0,45,24,0] >;

Q8×C23 in GAP, Magma, Sage, TeX

Q_8\times C_{23}
% in TeX

G:=Group("Q8xC23");
// GroupNames label

G:=SmallGroup(184,10);
// by ID

G=gap.SmallGroup(184,10);
# by ID

G:=PCGroup([4,-2,-2,-23,-2,368,753,373]);
// Polycyclic

G:=Group<a,b,c|a^23=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q8×C23 in TeX

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