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G = C40.31C23order 320 = 26·5

24th non-split extension by C40 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.9D10, C40.33D4, C20.52D8, C40.31C23, Q16.10D10, Dic20.14C22, D8.D56C2, C4○D8.3D5, C5⋊Q326C2, (C2×C10).11D8, C10.70(C2×D8), (C2×C8).99D10, C8.8(C5⋊D4), C55(Q32⋊C2), C20.4C87C2, C4.25(D4⋊D5), C20.193(C2×D4), (C2×C20).187D4, (C5×D8).9C22, C8.37(C22×D5), (C2×Dic20)⋊22C2, C52C16.4C22, C22.6(D4⋊D5), (C2×C40).105C22, (C5×Q16).10C22, C2.25(C2×D4⋊D5), (C5×C4○D8).4C2, C4.19(C2×C5⋊D4), (C2×C4).82(C5⋊D4), SmallGroup(320,822)

Series: Derived Chief Lower central Upper central

C1C40 — C40.31C23
C1C5C10C20C40Dic20C2×Dic20 — C40.31C23
C5C10C20C40 — C40.31C23
C1C2C2×C4C2×C8C4○D8

Generators and relations for C40.31C23
 G = < a,b,c,d | a40=c2=d2=1, b2=a20, bab-1=a-1, ac=ca, dad=a31, bc=cb, dbd=a25b, dcd=a20c >

Subgroups: 302 in 82 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C10, C10 [×2], C16 [×2], C2×C8, D8, SD16, Q16, Q16 [×3], C2×Q8, C4○D4, Dic5 [×2], C20 [×2], C20, C2×C10, C2×C10, M5(2), SD32 [×2], Q32 [×2], C2×Q16, C4○D8, C40 [×2], Dic10 [×3], C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C5×Q8, Q32⋊C2, C52C16 [×2], Dic20 [×2], Dic20, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×Dic10, C5×C4○D4, C20.4C8, D8.D5 [×2], C5⋊Q32 [×2], C2×Dic20, C5×C4○D8, C40.31C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C5⋊D4 [×2], C22×D5, Q32⋊C2, D4⋊D5 [×2], C2×C5⋊D4, C2×D4⋊D5, C40.31C23

Smallest permutation representation of C40.31C23
On 160 points
Generators in S160
(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 89 21 109)(2 88 22 108)(3 87 23 107)(4 86 24 106)(5 85 25 105)(6 84 26 104)(7 83 27 103)(8 82 28 102)(9 81 29 101)(10 120 30 100)(11 119 31 99)(12 118 32 98)(13 117 33 97)(14 116 34 96)(15 115 35 95)(16 114 36 94)(17 113 37 93)(18 112 38 92)(19 111 39 91)(20 110 40 90)(41 127 61 147)(42 126 62 146)(43 125 63 145)(44 124 64 144)(45 123 65 143)(46 122 66 142)(47 121 67 141)(48 160 68 140)(49 159 69 139)(50 158 70 138)(51 157 71 137)(52 156 72 136)(53 155 73 135)(54 154 74 134)(55 153 75 133)(56 152 76 132)(57 151 77 131)(58 150 78 130)(59 149 79 129)(60 148 80 128)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)(81 134)(82 135)(83 136)(84 137)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 151)(99 152)(100 153)(101 154)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 131)(119 132)(120 133)
(2 32)(3 23)(4 14)(6 36)(7 27)(8 18)(10 40)(11 31)(12 22)(15 35)(16 26)(19 39)(20 30)(24 34)(28 38)(41 71)(42 62)(43 53)(45 75)(46 66)(47 57)(49 79)(50 70)(51 61)(54 74)(55 65)(58 78)(59 69)(63 73)(67 77)(81 96)(82 87)(83 118)(84 109)(85 100)(86 91)(88 113)(89 104)(90 95)(92 117)(93 108)(94 99)(97 112)(98 103)(101 116)(102 107)(105 120)(106 111)(110 115)(114 119)(121 126)(122 157)(123 148)(124 139)(125 130)(127 152)(128 143)(129 134)(131 156)(132 147)(133 138)(135 160)(136 151)(137 142)(140 155)(141 146)(144 159)(145 150)(149 154)(153 158)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B8C10A10B10C10D10E10F10G10H16A16B16C16D20A20B20C20D20E20F20G20H20I20J40A···40H
order122244444558881010101010101010161616162020202020202020202040···40
size1128228404022224224488882020202022224488884···4

44 irreducible representations

dim11111122222222224444
type++++++++++++++-++-
imageC1C2C2C2C2C2D4D4D5D8D8D10D10D10C5⋊D4C5⋊D4Q32⋊C2D4⋊D5D4⋊D5C40.31C23
kernelC40.31C23C20.4C8D8.D5C5⋊Q32C2×Dic20C5×C4○D8C40C2×C20C4○D8C20C2×C10C2×C8D8Q16C8C2×C4C5C4C22C1
# reps11221111222222442228

Matrix representation of C40.31C23 in GL6(𝔽241)

190510000
1902400000
001123000
00111100
000011230
00001111
,
73920000
2251680000
0021824023946
0024023462
0023946218240
0046224023
,
24000000
02400000
000010
000001
001000
000100
,
24000000
02400000
001000
00024000
00002400
000001

G:=sub<GL(6,GF(241))| [190,190,0,0,0,0,51,240,0,0,0,0,0,0,11,11,0,0,0,0,230,11,0,0,0,0,0,0,11,11,0,0,0,0,230,11],[73,225,0,0,0,0,92,168,0,0,0,0,0,0,218,240,239,46,0,0,240,23,46,2,0,0,239,46,218,240,0,0,46,2,240,23],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1] >;

C40.31C23 in GAP, Magma, Sage, TeX

C_{40}._{31}C_2^3
% in TeX

G:=Group("C40.31C2^3");
// GroupNames label

G:=SmallGroup(320,822);
// by ID

G=gap.SmallGroup(320,822);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,387,675,185,192,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^40=c^2=d^2=1,b^2=a^20,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^31,b*c=c*b,d*b*d=a^25*b,d*c*d=a^20*c>;
// generators/relations

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