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G = C23.34D20order 320 = 26·5

5th non-split extension by C23 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.34D20, C406C46C2, (C2×C4).29D20, (C2×C20).40D4, C22⋊C8.6D5, (C2×C8).107D10, C10.6(C2×SD16), C20.44D49C2, (C2×C10).13SD16, (C22×C10).48D4, (C22×C4).75D10, C20.279(C4○D4), (C2×C20).738C23, (C2×C40).118C22, C20.48D4.2C2, C22.101(C2×D20), C10.7(C8.C22), C22.8(C40⋊C2), C51(C23.47D4), C4.103(D42D5), C2.10(C8.D10), C4⋊Dic5.268C22, (C22×C20).48C22, (C2×Dic10).12C22, C10.14(C22.D4), C2.10(C22.D20), C2.9(C2×C40⋊C2), (C5×C22⋊C8).8C2, (C2×C10).121(C2×D4), (C2×C4⋊Dic5).12C2, (C2×C4).683(C22×D5), SmallGroup(320,348)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C23.34D20
C1C5C10C20C2×C20C4⋊Dic5C2×C4⋊Dic5 — C23.34D20
C5C10C2×C20 — C23.34D20
C1C22C22×C4C22⋊C8

Generators and relations for C23.34D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=cb=bc, dad-1=eae-1=ab=ba, ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d19 >

Subgroups: 398 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×5], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C22⋊Q8, C40 [×2], Dic10 [×2], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×2], C22×C10, C23.47D4, C10.D4, C4⋊Dic5, C4⋊Dic5 [×2], C4⋊Dic5, C23.D5, C2×C40 [×2], C2×Dic10, C22×Dic5, C22×C20, C20.44D4 [×2], C406C4 [×2], C5×C22⋊C8, C20.48D4, C2×C4⋊Dic5, C23.34D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×SD16, C8.C22, D20 [×2], C22×D5, C23.47D4, C40⋊C2 [×2], C2×D20, D42D5 [×2], C22.D20, C2×C40⋊C2, C8.D10, C23.34D20

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444455888810···101010101020···202020202040···40
size1111222242020202040402244442···244442···244444···4

59 irreducible representations

dim1111112222222222444
type+++++++++++++---
imageC1C2C2C2C2C2D4D4D5C4○D4SD16D10D10D20D20C40⋊C2C8.C22D42D5C8.D10
kernelC23.34D20C20.44D4C406C4C5×C22⋊C8C20.48D4C2×C4⋊Dic5C2×C20C22×C10C22⋊C8C20C2×C10C2×C8C22×C4C2×C4C23C22C10C4C2
# reps12211111244424416144

Matrix representation of C23.34D20 in GL4(𝔽41) generated by

1000
04000
0010
0001
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
0100
1000
002316
002512
,
0900
9000
002927
002512
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,23,25,0,0,16,12],[0,9,0,0,9,0,0,0,0,0,29,25,0,0,27,12] >;

C23.34D20 in GAP, Magma, Sage, TeX

C_2^3._{34}D_{20}
% in TeX

G:=Group("C2^3.34D20");
// GroupNames label

G:=SmallGroup(320,348);
// by ID

G=gap.SmallGroup(320,348);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,254,219,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*b=b*c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*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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