Copied to
clipboard

G = D4.3Dic10order 320 = 26·5

The non-split extension by D4 of Dic10 acting via Dic10/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.3Dic10, C42.46D10, (C4×D4).4D5, C56(D4.Q8), (C5×D4).3Q8, C203C819C2, (D4×C20).4C2, (C2×C20).59D4, C20.27(C2×Q8), C4⋊C4.240D10, (C2×D4).187D10, C20.47(C4○D4), C4.61(C4○D20), C10.87(C4○D8), C10.D831C2, (C4×C20).80C22, C20.Q831C2, C20.6Q811C2, C4.11(C2×Dic10), D4⋊Dic5.9C2, C10.86(C8⋊C22), (C2×C20).334C23, C10.63(C22⋊Q8), C2.8(D4.D10), (D4×C10).229C22, C4⋊Dic5.138C22, C2.10(D4.8D10), C2.14(C20.48D4), (C2×C10).465(C2×D4), (C2×C4).216(C5⋊D4), (C5×C4⋊C4).271C22, (C2×C52C8).91C22, (C2×C4).434(C22×D5), C22.148(C2×C5⋊D4), SmallGroup(320,636)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.3Dic10
C1C5C10C20C2×C20C4⋊Dic5C20.6Q8 — D4.3Dic10
C5C10C2×C20 — D4.3Dic10
C1C22C42C4×D4

Generators and relations for D4.3Dic10
 G = < a,b,c,d | a4=b2=c20=1, d2=a2c10, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 310 in 102 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C2×D4, Dic5 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×4], D4⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8 [×2], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×3], C5×D4 [×2], C5×D4, C22×C10, D4.Q8, C2×C52C8 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C203C8, C10.D8, C20.Q8, D4⋊Dic5 [×2], C20.6Q8, D4×C20, D4.3Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8⋊C22, Dic10 [×2], C5⋊D4 [×2], C22×D5, D4.Q8, C2×Dic10, C4○D20, C2×C5⋊D4, C20.48D4, D4.D10, D4.8D10, D4.3Dic10

Smallest permutation representation of D4.3Dic10
On 160 points
Generators in S160
(1 58 38 148)(2 59 39 149)(3 60 40 150)(4 41 21 151)(5 42 22 152)(6 43 23 153)(7 44 24 154)(8 45 25 155)(9 46 26 156)(10 47 27 157)(11 48 28 158)(12 49 29 159)(13 50 30 160)(14 51 31 141)(15 52 32 142)(16 53 33 143)(17 54 34 144)(18 55 35 145)(19 56 36 146)(20 57 37 147)(61 118 134 88)(62 119 135 89)(63 120 136 90)(64 101 137 91)(65 102 138 92)(66 103 139 93)(67 104 140 94)(68 105 121 95)(69 106 122 96)(70 107 123 97)(71 108 124 98)(72 109 125 99)(73 110 126 100)(74 111 127 81)(75 112 128 82)(76 113 129 83)(77 114 130 84)(78 115 131 85)(79 116 132 86)(80 117 133 87)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 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)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(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)
(1 132 28 69)(2 78 29 121)(3 130 30 67)(4 76 31 139)(5 128 32 65)(6 74 33 137)(7 126 34 63)(8 72 35 135)(9 124 36 61)(10 70 37 133)(11 122 38 79)(12 68 39 131)(13 140 40 77)(14 66 21 129)(15 138 22 75)(16 64 23 127)(17 136 24 73)(18 62 25 125)(19 134 26 71)(20 80 27 123)(41 83 141 103)(42 112 142 92)(43 81 143 101)(44 110 144 90)(45 99 145 119)(46 108 146 88)(47 97 147 117)(48 106 148 86)(49 95 149 115)(50 104 150 84)(51 93 151 113)(52 102 152 82)(53 91 153 111)(54 120 154 100)(55 89 155 109)(56 118 156 98)(57 87 157 107)(58 116 158 96)(59 85 159 105)(60 114 160 94)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G···10N20A···20H20I···20X
order12222244444444455888810···1010···1020···2020···20
size1111442222444404022202020202···24···42···24···4

59 irreducible representations

dim111111122222222222444
type++++++++-++++-+
imageC1C2C2C2C2C2C2D4Q8D5C4○D4D10D10D10C4○D8C5⋊D4Dic10C4○D20C8⋊C22D4.D10D4.8D10
kernelD4.3Dic10C203C8C10.D8C20.Q8D4⋊Dic5C20.6Q8D4×C20C2×C20C5×D4C4×D4C20C42C4⋊C4C2×D4C10C2×C4D4C4C10C2C2
# reps111121122222224888144

Matrix representation of D4.3Dic10 in GL4(𝔽41) generated by

0100
40000
0010
0001
,
0100
1000
00400
00040
,
32000
03200
002839
00216
,
122900
292900
002739
003714
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,28,2,0,0,39,16],[12,29,0,0,29,29,0,0,0,0,27,37,0,0,39,14] >;

D4.3Dic10 in GAP, Magma, Sage, TeX

D_4._3{\rm Dic}_{10}
% in TeX

G:=Group("D4.3Dic10");
// GroupNames label

G:=SmallGroup(320,636);
// by ID

G=gap.SmallGroup(320,636);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,1123,297,136,12550]);
// Polycyclic

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

׿
×
𝔽