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G = C23.22D20order 320 = 26·5

1st non-split extension by C23 of D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.22D20, (C2×C40)⋊20C4, (C2×C8)⋊6Dic5, C406C429C2, C405C429C2, C20.78(C4⋊C4), (C2×C20).61Q8, C20.75(C2×Q8), C40.114(C2×C4), (C2×C4).169D20, (C2×C20).401D4, (C2×C8).307D10, C8.17(C2×Dic5), (C22×C8).11D5, C10.15(C4○D8), (C22×C40).17C2, C4.23(C4⋊Dic5), (C2×C4).50Dic10, C4.41(C2×Dic10), C22.52(C2×D20), C2.4(D407C2), (C2×C20).765C23, C20.230(C22×C4), (C2×C40).392C22, (C22×C4).425D10, (C22×C10).137D4, C55(C23.25D4), C4.25(C22×Dic5), C4⋊Dic5.281C22, C22.13(C4⋊Dic5), (C22×C20).539C22, C23.21D10.5C2, C10.69(C2×C4⋊C4), C2.12(C2×C4⋊Dic5), (C2×C10).79(C4⋊C4), (C2×C20).479(C2×C4), (C2×C10).155(C2×D4), (C2×C4).83(C2×Dic5), (C2×C4).712(C22×D5), SmallGroup(320,733)

Series: Derived Chief Lower central Upper central

C1C20 — C23.22D20
C1C5C10C2×C10C2×C20C4⋊Dic5C23.21D10 — C23.22D20
C5C10C20 — C23.22D20
C1C2×C4C22×C4C22×C8

Generators and relations for C23.22D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d19 >

Subgroups: 334 in 114 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×4], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22×C8, C40 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C23.25D4, C4×Dic5 [×2], C4⋊Dic5 [×4], C23.D5 [×2], C2×C40 [×2], C2×C40 [×4], C22×C20, C406C4 [×2], C405C4 [×2], C23.21D10 [×2], C22×C40, C23.22D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], C2×C4⋊C4, C4○D8 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, C23.25D4, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, D407C2 [×2], C2×C4⋊Dic5, C23.22D20

Smallest permutation representation of C23.22D20
On 160 points
Generators in S160
(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 138)(42 139)(43 140)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 153)(57 154)(58 155)(59 156)(60 157)(61 158)(62 159)(63 160)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 137)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(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)
(1 56 109 133)(2 75 110 152)(3 54 111 131)(4 73 112 150)(5 52 113 129)(6 71 114 148)(7 50 115 127)(8 69 116 146)(9 48 117 125)(10 67 118 144)(11 46 119 123)(12 65 120 142)(13 44 81 121)(14 63 82 140)(15 42 83 159)(16 61 84 138)(17 80 85 157)(18 59 86 136)(19 78 87 155)(20 57 88 134)(21 76 89 153)(22 55 90 132)(23 74 91 151)(24 53 92 130)(25 72 93 149)(26 51 94 128)(27 70 95 147)(28 49 96 126)(29 68 97 145)(30 47 98 124)(31 66 99 143)(32 45 100 122)(33 64 101 141)(34 43 102 160)(35 62 103 139)(36 41 104 158)(37 60 105 137)(38 79 106 156)(39 58 107 135)(40 77 108 154)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N5A5B8A···8H10A···10N20A···20P40A···40AF
order1222224444444···4558···810···1020···2040···40
size11112211112220···20222···22···22···22···2

92 irreducible representations

dim111111222222222222
type++++++-++-++-++
imageC1C2C2C2C2C4D4Q8D4D5Dic5D10D10C4○D8Dic10D20D20D407C2
kernelC23.22D20C406C4C405C4C23.21D10C22×C40C2×C40C2×C20C2×C20C22×C10C22×C8C2×C8C2×C8C22×C4C10C2×C4C2×C4C23C2
# reps1222181212842884432

Matrix representation of C23.22D20 in GL3(𝔽41) generated by

4000
010
0040
,
4000
010
001
,
100
0400
0040
,
4000
0190
0028
,
3200
0028
0190
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[40,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,40],[40,0,0,0,19,0,0,0,28],[32,0,0,0,0,19,0,28,0] >;

C23.22D20 in GAP, Magma, Sage, TeX

C_2^3._{22}D_{20}
% in TeX

G:=Group("C2^3.22D20");
// GroupNames label

G:=SmallGroup(320,733);
// by ID

G=gap.SmallGroup(320,733);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,100,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^19>;
// generators/relations

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