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## G = Q8×D23order 368 = 24·23

### Direct product of Q8 and D23

Aliases: Q8×D23, C4.6D46, Dic464C2, C46.7C23, C92.6C22, D46.5C22, Dic23.3C22, C232(C2×Q8), (Q8×C23)⋊2C2, (C4×D23).1C2, C2.8(C22×D23), SmallGroup(368,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C46 — Q8×D23
 Chief series C1 — C23 — C46 — D46 — C4×D23 — Q8×D23
 Lower central C23 — C46 — Q8×D23
 Upper central C1 — C2 — Q8

Generators and relations for Q8×D23
G = < a,b,c,d | a4=c23=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of Q8×D23
On 184 points
Generators in S184
(1 80 39 58)(2 81 40 59)(3 82 41 60)(4 83 42 61)(5 84 43 62)(6 85 44 63)(7 86 45 64)(8 87 46 65)(9 88 24 66)(10 89 25 67)(11 90 26 68)(12 91 27 69)(13 92 28 47)(14 70 29 48)(15 71 30 49)(16 72 31 50)(17 73 32 51)(18 74 33 52)(19 75 34 53)(20 76 35 54)(21 77 36 55)(22 78 37 56)(23 79 38 57)(93 141 120 174)(94 142 121 175)(95 143 122 176)(96 144 123 177)(97 145 124 178)(98 146 125 179)(99 147 126 180)(100 148 127 181)(101 149 128 182)(102 150 129 183)(103 151 130 184)(104 152 131 162)(105 153 132 163)(106 154 133 164)(107 155 134 165)(108 156 135 166)(109 157 136 167)(110 158 137 168)(111 159 138 169)(112 160 116 170)(113 161 117 171)(114 139 118 172)(115 140 119 173)
(1 126 39 99)(2 127 40 100)(3 128 41 101)(4 129 42 102)(5 130 43 103)(6 131 44 104)(7 132 45 105)(8 133 46 106)(9 134 24 107)(10 135 25 108)(11 136 26 109)(12 137 27 110)(13 138 28 111)(14 116 29 112)(15 117 30 113)(16 118 31 114)(17 119 32 115)(18 120 33 93)(19 121 34 94)(20 122 35 95)(21 123 36 96)(22 124 37 97)(23 125 38 98)(47 169 92 159)(48 170 70 160)(49 171 71 161)(50 172 72 139)(51 173 73 140)(52 174 74 141)(53 175 75 142)(54 176 76 143)(55 177 77 144)(56 178 78 145)(57 179 79 146)(58 180 80 147)(59 181 81 148)(60 182 82 149)(61 183 83 150)(62 184 84 151)(63 162 85 152)(64 163 86 153)(65 164 87 154)(66 165 88 155)(67 166 89 156)(68 167 90 157)(69 168 91 158)
(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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 92)(69 91)(93 131)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 123)(102 122)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 138)(110 137)(111 136)(112 135)(113 134)(114 133)(115 132)(139 164)(140 163)(141 162)(142 184)(143 183)(144 182)(145 181)(146 180)(147 179)(148 178)(149 177)(150 176)(151 175)(152 174)(153 173)(154 172)(155 171)(156 170)(157 169)(158 168)(159 167)(160 166)(161 165)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 23A ··· 23K 46A ··· 46K 92A ··· 92AG order 1 2 2 2 4 4 4 4 4 4 23 ··· 23 46 ··· 46 92 ··· 92 size 1 1 23 23 2 2 2 46 46 46 2 ··· 2 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 2 2 2 4 type + + + + - + + - image C1 C2 C2 C2 Q8 D23 D46 Q8×D23 kernel Q8×D23 Dic46 C4×D23 Q8×C23 D23 Q8 C4 C1 # reps 1 3 3 1 2 11 33 11

Matrix representation of Q8×D23 in GL4(𝔽277) generated by

 1 0 0 0 0 1 0 0 0 0 1 168 0 0 122 276
,
 276 0 0 0 0 276 0 0 0 0 228 145 0 0 14 49
,
 22 1 0 0 94 168 0 0 0 0 1 0 0 0 0 1
,
 81 188 0 0 248 196 0 0 0 0 276 0 0 0 0 276
G:=sub<GL(4,GF(277))| [1,0,0,0,0,1,0,0,0,0,1,122,0,0,168,276],[276,0,0,0,0,276,0,0,0,0,228,14,0,0,145,49],[22,94,0,0,1,168,0,0,0,0,1,0,0,0,0,1],[81,248,0,0,188,196,0,0,0,0,276,0,0,0,0,276] >;

Q8×D23 in GAP, Magma, Sage, TeX

Q_8\times D_{23}
% in TeX

G:=Group("Q8xD23");
// GroupNames label

G:=SmallGroup(368,33);
// by ID

G=gap.SmallGroup(368,33);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-23,46,97,42,8804]);
// Polycyclic

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

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