Copied to
clipboard

G = D20.30D4order 320 = 26·5

13rd non-split extension by D20 of D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.30D4, C20.13C24, C40.42C23, Q16.11D10, D20.8C23, Dic10.30D4, D40.13C22, Dic10.8C23, Dic20.15C22, C52(Q8○D8), (D5×Q16)⋊6C2, C4.77(D4×D5), Q8⋊D5.C22, (C10×Q16)⋊3C2, (C2×Q16)⋊12D5, C20.88(C2×D4), C5⋊D4.10D4, D407C24C2, Q8.D106C2, Q16⋊D55C2, D10.51(C2×D4), (C2×C8).105D10, C52C8.5C23, (C2×Q8).91D10, (C8×D5).7C22, (C4×D5).7C23, C8.14(C22×D5), C4.13(C23×D5), C22.22(D4×D5), D20.3C43C2, Q8.7(C22×D5), (C5×Q8).7C23, (Q8×D5).1C22, C20.C238C2, (C2×C40).35C22, Dic5.57(C2×D4), C8⋊D5.3C22, C40⋊C2.3C22, C5⋊Q16.1C22, (C2×C20).530C23, Q8.10D104C2, C4○D20.53C22, C10.114(C22×D4), Q82D5.1C22, (C5×Q16).11C22, (Q8×C10).152C22, C4.Dic5.48C22, C2.87(C2×D4×D5), (C2×C10).403(C2×D4), (C2×C4).231(C22×D5), SmallGroup(320,1438)

Series: Derived Chief Lower central Upper central

C1C20 — D20.30D4
C1C5C10C20C4×D5C4○D20Q8.10D10 — D20.30D4
C5C10C20 — D20.30D4
C1C2C2×C4C2×Q16

Generators and relations for D20.30D4
 G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a10c3 >

Subgroups: 886 in 248 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×11], Q8 [×4], Q8 [×9], D5 [×4], C10, C10, C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×6], Q16 [×4], Q16 [×5], C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×13], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×2], C2×C10, C8○D4, C2×Q16, C2×Q16 [×2], C4○D8 [×3], C8.C22 [×6], 2- 1+4 [×2], C52C8 [×2], C40 [×2], Dic10, Dic10 [×2], Dic10 [×4], C4×D5 [×2], C4×D5 [×10], D20, D20 [×2], D20 [×4], C5⋊D4 [×2], C5⋊D4 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], Q8○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C4.Dic5, Q8⋊D5 [×4], C5⋊Q16 [×4], C2×C40, C5×Q16 [×4], C4○D20, C4○D20 [×2], C4○D20 [×4], Q8×D5 [×4], Q8×D5 [×2], Q82D5 [×4], Q82D5 [×2], Q8×C10 [×2], D20.3C4, D407C2, D5×Q16 [×2], Q16⋊D5 [×4], Q8.D10 [×2], C20.C23 [×2], C10×Q16, Q8.10D10 [×2], D20.30D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D20.30D4

Smallest permutation representation of D20.30D4
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 25)(2 24)(3 23)(4 22)(5 21)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 160)(48 159)(49 158)(50 157)(51 156)(52 155)(53 154)(54 153)(55 152)(56 151)(57 150)(58 149)(59 148)(60 147)(61 135)(62 134)(63 133)(64 132)(65 131)(66 130)(67 129)(68 128)(69 127)(70 126)(71 125)(72 124)(73 123)(74 122)(75 121)(76 140)(77 139)(78 138)(79 137)(80 136)(81 102)(82 101)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)
(1 80 93 52 11 70 83 42)(2 61 94 53 12 71 84 43)(3 62 95 54 13 72 85 44)(4 63 96 55 14 73 86 45)(5 64 97 56 15 74 87 46)(6 65 98 57 16 75 88 47)(7 66 99 58 17 76 89 48)(8 67 100 59 18 77 90 49)(9 68 81 60 19 78 91 50)(10 69 82 41 20 79 92 51)(21 132 106 151 31 122 116 141)(22 133 107 152 32 123 117 142)(23 134 108 153 33 124 118 143)(24 135 109 154 34 125 119 144)(25 136 110 155 35 126 120 145)(26 137 111 156 36 127 101 146)(27 138 112 157 37 128 102 147)(28 139 113 158 38 129 103 148)(29 140 114 159 39 130 104 149)(30 121 115 160 40 131 105 150)
(1 106 11 116)(2 117 12 107)(3 108 13 118)(4 119 14 109)(5 110 15 120)(6 101 16 111)(7 112 17 102)(8 103 18 113)(9 114 19 104)(10 105 20 115)(21 83 31 93)(22 94 32 84)(23 85 33 95)(24 96 34 86)(25 87 35 97)(26 98 36 88)(27 89 37 99)(28 100 38 90)(29 91 39 81)(30 82 40 92)(41 160 51 150)(42 151 52 141)(43 142 53 152)(44 153 54 143)(45 144 55 154)(46 155 56 145)(47 146 57 156)(48 157 58 147)(49 148 59 158)(50 159 60 149)(61 123 71 133)(62 134 72 124)(63 125 73 135)(64 136 74 126)(65 127 75 137)(66 138 76 128)(67 129 77 139)(68 140 78 130)(69 131 79 121)(70 122 80 132)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444444558888810···102020202020···2040···40
size11210102020224444101020202222420202···244448···84···4

50 irreducible representations

dim11111111122222224444
type++++++++++++++++-++
imageC1C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10Q8○D8D4×D5D4×D5D20.30D4
kernelD20.30D4D20.3C4D407C2D5×Q16Q16⋊D5Q8.D10C20.C23C10×Q16Q8.10D10Dic10D20C5⋊D4C2×Q16C2×C8Q16C2×Q8C5C4C22C1
# reps11124221211222842228

Matrix representation of D20.30D4 in GL4(𝔽41) generated by

00623
001821
351800
232000
,
817015
2433150
026817
2602433
,
120290
012029
120120
012012
,
150178
0153324
178260
3324026
G:=sub<GL(4,GF(41))| [0,0,35,23,0,0,18,20,6,18,0,0,23,21,0,0],[8,24,0,26,17,33,26,0,0,15,8,24,15,0,17,33],[12,0,12,0,0,12,0,12,29,0,12,0,0,29,0,12],[15,0,17,33,0,15,8,24,17,33,26,0,8,24,0,26] >;

D20.30D4 in GAP, Magma, Sage, TeX

D_{20}._{30}D_4
% in TeX

G:=Group("D20.30D4");
// GroupNames label

G:=SmallGroup(320,1438);
// by ID

G=gap.SmallGroup(320,1438);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=a^10*c^3>;
// generators/relations

׿
×
𝔽