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G = C23.10D20order 320 = 26·5

3rd non-split extension by C23 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.10D20, C405C43C2, C406C47C2, (C2×C8).2D10, C22⋊C8.5D5, C10.7(C4○D8), (C2×C20).239D4, (C2×C4).117D20, (C2×C40).2C22, C20.44D45C2, (C22×C4).77D10, (C22×C10).50D4, C20.281(C4○D4), C2.9(D407C2), (C2×C20).740C23, C20.48D4.8C2, C22.103(C2×D20), C10.8(C8.C22), C51(C23.20D4), C4.105(D42D5), C2.11(C8.D10), C4⋊Dic5.269C22, (C22×C20).92C22, (C2×Dic10).14C22, C23.21D10.3C2, C10.16(C22.D4), C2.12(C22.D20), (C5×C22⋊C8).7C2, (C2×C10).123(C2×D4), (C2×C4).685(C22×D5), SmallGroup(320,350)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C23.10D20
C1C5C10C20C2×C20C4⋊Dic5C23.21D10 — C23.10D20
C5C10C2×C20 — C23.10D20
C1C22C22×C4C22⋊C8

Generators and relations for C23.10D20
 G = < a,b,c,d,e | a2=b2=c2=1, d20=e2=c, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd19 >

Subgroups: 350 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C22×C4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×3], C22⋊C8, Q8⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C40 [×2], Dic10 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×C10, C23.20D4, C4×Dic5, C10.D4, C4⋊Dic5 [×3], C23.D5 [×2], C2×C40 [×2], C2×Dic10, C22×C20, C20.44D4 [×2], C406C4, C405C4, C5×C22⋊C8, C20.48D4, C23.21D10, C23.10D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8.C22, D20 [×2], C22×D5, C23.20D4, C2×D20, D42D5 [×2], C22.D20, D407C2, C8.D10, C23.10D20

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222444444444455888810···101010101020···202020202040···40
size1111422222020202040402244442···244442···244444···4

59 irreducible representations

dim11111112222222222444
type++++++++++++++---
imageC1C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D8D20D20D407C2C8.C22D42D5C8.D10
kernelC23.10D20C20.44D4C406C4C405C4C5×C22⋊C8C20.48D4C23.21D10C2×C20C22×C10C22⋊C8C20C2×C8C22×C4C10C2×C4C23C2C10C4C2
# reps121111111244244416144

Matrix representation of C23.10D20 in GL4(𝔽41) generated by

1000
04000
0010
004040
,
1000
0100
00400
00040
,
40000
04000
0010
0001
,
24000
02900
004039
0011
,
02900
24000
00918
003232
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,40,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[24,0,0,0,0,29,0,0,0,0,40,1,0,0,39,1],[0,24,0,0,29,0,0,0,0,0,9,32,0,0,18,32] >;

C23.10D20 in GAP, Magma, Sage, TeX

C_2^3._{10}D_{20}
% in TeX

G:=Group("C2^3.10D20");
// GroupNames label

G:=SmallGroup(320,350);
// by ID

G=gap.SmallGroup(320,350);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,254,219,142,1123,136,12550]);
// Polycyclic

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

׿
×
𝔽