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G = C23.35D20order 320 = 26·5

6th non-split extension by C23 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.35D20, C22.3Dic20, C405C42C2, (C2×C8).1D10, (C2×C4).30D20, (C2×C20).41D4, C22⋊C8.3D5, (C2×C10).4Q16, C10.5(C2×Q16), (C2×C40).1C22, C2.7(C2×Dic20), C20.44D44C2, C10.7(C8⋊C22), (C22×C10).49D4, (C22×C4).76D10, C20.280(C4○D4), C2.10(C8⋊D10), (C2×C20).739C23, C20.48D4.3C2, C22.102(C2×D20), C51(C23.48D4), C4⋊Dic5.11C22, C4.104(D42D5), (C22×C20).49C22, (C2×Dic10).13C22, C10.15(C22.D4), C2.11(C22.D20), (C5×C22⋊C8).5C2, (C2×C10).122(C2×D4), (C2×C4⋊Dic5).13C2, (C2×C4).684(C22×D5), SmallGroup(320,349)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C23.35D20
C1C5C10C20C2×C20C4⋊Dic5C2×C4⋊Dic5 — C23.35D20
C5C10C2×C20 — C23.35D20
C1C22C22×C4C22⋊C8

Generators and relations for C23.35D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=e2=c, ab=ba, ac=ca, dad-1=eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >

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], C2.D8 [×2], C2×C4⋊C4, C22⋊Q8, C40 [×2], Dic10 [×2], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×2], C22×C10, C23.48D4, C10.D4, C4⋊Dic5, C4⋊Dic5 [×2], C4⋊Dic5, C23.D5, C2×C40 [×2], C2×Dic10, C22×Dic5, C22×C20, C20.44D4 [×2], C405C4 [×2], C5×C22⋊C8, C20.48D4, C2×C4⋊Dic5, C23.35D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, Q16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×Q16, C8⋊C22, D20 [×2], C22×D5, C23.48D4, Dic20 [×2], C2×D20, D42D5 [×2], C22.D20, C2×Dic20, C8⋊D10, C23.35D20

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

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

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

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

59 conjugacy classes

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

59 irreducible representations

dim1111112222222222444
type+++++++++-++++-+-+
imageC1C2C2C2C2C2D4D4D5C4○D4Q16D10D10D20D20Dic20C8⋊C22D42D5C8⋊D10
kernelC23.35D20C20.44D4C405C4C5×C22⋊C8C20.48D4C2×C4⋊Dic5C2×C20C22×C10C22⋊C8C20C2×C10C2×C8C22×C4C2×C4C23C22C10C4C2
# reps12211111244424416144

Matrix representation of C23.35D20 in GL6(𝔽41)

100000
010000
001000
000100
000010
00004040
,
4000000
0400000
001000
000100
0000400
0000040
,
4000000
0400000
001000
000100
000010
000001
,
12290000
12120000
00273000
00113200
00004039
000011
,
2130000
3200000
0073400
0013400
0000918
00003232

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,29,12,0,0,0,0,0,0,27,11,0,0,0,0,30,32,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[21,3,0,0,0,0,3,20,0,0,0,0,0,0,7,1,0,0,0,0,34,34,0,0,0,0,0,0,9,32,0,0,0,0,18,32] >;

C23.35D20 in GAP, Magma, Sage, TeX

C_2^3._{35}D_{20}
% in TeX

G:=Group("C2^3.35D20");
// GroupNames label

G:=SmallGroup(320,349);
// by ID

G=gap.SmallGroup(320,349);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,336,254,219,310,1123,136,12550]);
// Polycyclic

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

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×
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