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## G = C4○D4×C23order 368 = 24·23

### Direct product of C23 and C4○D4

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

Aliases: C4○D4×C23, D42C46, Q82C46, C46.13C23, C92.21C22, (C2×C4)⋊3C46, (C2×C92)⋊7C2, (D4×C23)⋊5C2, C4.5(C2×C46), (Q8×C23)⋊5C2, C22.(C2×C46), C2.3(C22×C46), (C2×C46).2C22, SmallGroup(368,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C23
 Chief series C1 — C2 — C46 — C2×C46 — D4×C23 — C4○D4×C23
 Lower central C1 — C2 — C4○D4×C23
 Upper central C1 — C92 — C4○D4×C23

Generators and relations for C4○D4×C23
G = < a,b,c,d | a23=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Smallest permutation representation of C4○D4×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 95 161 32)(2 96 139 33)(3 97 140 34)(4 98 141 35)(5 99 142 36)(6 100 143 37)(7 101 144 38)(8 102 145 39)(9 103 146 40)(10 104 147 41)(11 105 148 42)(12 106 149 43)(13 107 150 44)(14 108 151 45)(15 109 152 46)(16 110 153 24)(17 111 154 25)(18 112 155 26)(19 113 156 27)(20 114 157 28)(21 115 158 29)(22 93 159 30)(23 94 160 31)(47 164 73 136)(48 165 74 137)(49 166 75 138)(50 167 76 116)(51 168 77 117)(52 169 78 118)(53 170 79 119)(54 171 80 120)(55 172 81 121)(56 173 82 122)(57 174 83 123)(58 175 84 124)(59 176 85 125)(60 177 86 126)(61 178 87 127)(62 179 88 128)(63 180 89 129)(64 181 90 130)(65 182 91 131)(66 183 92 132)(67 184 70 133)(68 162 71 134)(69 163 72 135)
(1 32 161 95)(2 33 139 96)(3 34 140 97)(4 35 141 98)(5 36 142 99)(6 37 143 100)(7 38 144 101)(8 39 145 102)(9 40 146 103)(10 41 147 104)(11 42 148 105)(12 43 149 106)(13 44 150 107)(14 45 151 108)(15 46 152 109)(16 24 153 110)(17 25 154 111)(18 26 155 112)(19 27 156 113)(20 28 157 114)(21 29 158 115)(22 30 159 93)(23 31 160 94)(47 164 73 136)(48 165 74 137)(49 166 75 138)(50 167 76 116)(51 168 77 117)(52 169 78 118)(53 170 79 119)(54 171 80 120)(55 172 81 121)(56 173 82 122)(57 174 83 123)(58 175 84 124)(59 176 85 125)(60 177 86 126)(61 178 87 127)(62 179 88 128)(63 180 89 129)(64 181 90 130)(65 182 91 131)(66 183 92 132)(67 184 70 133)(68 162 71 134)(69 163 72 135)
(1 177)(2 178)(3 179)(4 180)(5 181)(6 182)(7 183)(8 184)(9 162)(10 163)(11 164)(12 165)(13 166)(14 167)(15 168)(16 169)(17 170)(18 171)(19 172)(20 173)(21 174)(22 175)(23 176)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 47)(43 48)(44 49)(45 50)(46 51)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(91 100)(92 101)(116 151)(117 152)(118 153)(119 154)(120 155)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 139)(128 140)(129 141)(130 142)(131 143)(132 144)(133 145)(134 146)(135 147)(136 148)(137 149)(138 150)

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

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

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

230 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 23A ··· 23V 46A ··· 46V 46W ··· 46CJ 92A ··· 92AR 92AS ··· 92DF order 1 2 2 2 2 4 4 4 4 4 23 ··· 23 46 ··· 46 46 ··· 46 92 ··· 92 92 ··· 92 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

230 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C23 C46 C46 C46 C4○D4 C4○D4×C23 kernel C4○D4×C23 C2×C92 D4×C23 Q8×C23 C4○D4 C2×C4 D4 Q8 C23 C1 # reps 1 3 3 1 22 66 66 22 2 44

Matrix representation of C4○D4×C23 in GL2(𝔽277) generated by

 155 0 0 155
,
 60 0 0 60
,
 217 0 0 60
,
 0 60 217 0
G:=sub<GL(2,GF(277))| [155,0,0,155],[60,0,0,60],[217,0,0,60],[0,217,60,0] >;

C4○D4×C23 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{23}
% in TeX

G:=Group("C4oD4xC23");
// GroupNames label

G:=SmallGroup(368,40);
// by ID

G=gap.SmallGroup(368,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-23,-2,1861,702]);
// Polycyclic

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

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