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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

C1C2 — C4○D4×C23
C1C2C46C2×C46D4×C23 — C4○D4×C23
C1C2 — C4○D4×C23
C1C92 — 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 >

2C2
2C2
2C2
2C46
2C46
2C46

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

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

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

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

230 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E23A···23V46A···46V46W···46CJ92A···92AR92AS···92DF
order122224444423···2346···4646···4692···9292···92
size11222112221···11···12···21···12···2

230 irreducible representations

dim1111111122
type++++
imageC1C2C2C2C23C46C46C46C4○D4C4○D4×C23
kernelC4○D4×C23C2×C92D4×C23Q8×C23C4○D4C2×C4D4Q8C23C1
# reps133122666622244

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

1550
0155
,
600
060
,
2170
060
,
060
2170
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

Export

Subgroup lattice of C4○D4×C23 in TeX

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