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G = (D4×C10).24C4order 320 = 26·5

5th non-split extension by D4×C10 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (D4×C10).24C4, C20.451(C2×D4), (C2×C20).196D4, (Q8×C10).21C4, (C2×D4).8Dic5, (C2×Q8).7Dic5, C10.67(C8○D4), C20.55D431C2, C20.84(C22⋊C4), (C2×C20).872C23, (C22×C4).356D10, C2.9(D4.Dic5), C4.14(C23.D5), C23.18(C2×Dic5), C22.4(C23.D5), (C22×C20).376C22, C22.54(C22×Dic5), (C2×C4○D4).3D5, (C22×C52C8)⋊9C2, (C10×C4○D4).3C2, C4.142(C2×C5⋊D4), (C2×C20).297(C2×C4), C57((C22×C8)⋊C2), (C2×C4.Dic5)⋊20C2, (C2×C4).53(C2×Dic5), C2.19(C2×C23.D5), (C2×C4).200(C5⋊D4), C10.124(C2×C22⋊C4), (C2×C4).814(C22×D5), (C2×C10).90(C22⋊C4), (C2×C10).301(C22×C4), (C22×C10).142(C2×C4), (C2×C52C8).334C22, SmallGroup(320,861)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (D4×C10).24C4
C1C5C10C20C2×C20C2×C52C8C22×C52C8 — (D4×C10).24C4
C5C2×C10 — (D4×C10).24C4
C1C2×C4C2×C4○D4

Generators and relations for (D4×C10).24C4
 G = < a,b,c,d | a10=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, bd=db, dcd-1=a5b2c >

Subgroups: 350 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×8], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C10, C10 [×2], C10 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×8], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C52C8 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×C10, C22×C10 [×2], (C22×C8)⋊C2, C2×C52C8 [×4], C2×C52C8 [×2], C4.Dic5 [×2], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C20.55D4 [×4], C22×C52C8, C2×C4.Dic5, C10×C4○D4, (D4×C10).24C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C8○D4 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, (C22×C8)⋊C2, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4.Dic5 [×2], C2×C23.D5, (D4×C10).24C4

Smallest permutation representation of (D4×C10).24C4
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 78 23 86)(2 79 24 87)(3 80 25 88)(4 71 26 89)(5 72 27 90)(6 73 28 81)(7 74 29 82)(8 75 30 83)(9 76 21 84)(10 77 22 85)(11 106 158 98)(12 107 159 99)(13 108 160 100)(14 109 151 91)(15 110 152 92)(16 101 153 93)(17 102 154 94)(18 103 155 95)(19 104 156 96)(20 105 157 97)(31 51 48 68)(32 52 49 69)(33 53 50 70)(34 54 41 61)(35 55 42 62)(36 56 43 63)(37 57 44 64)(38 58 45 65)(39 59 46 66)(40 60 47 67)(111 131 129 149)(112 132 130 150)(113 133 121 141)(114 134 122 142)(115 135 123 143)(116 136 124 144)(117 137 125 145)(118 138 126 146)(119 139 127 147)(120 140 128 148)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 71)(10 72)(11 106)(12 107)(13 108)(14 109)(15 110)(16 101)(17 102)(18 103)(19 104)(20 105)(21 89)(22 90)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 56)(32 57)(33 58)(34 59)(35 60)(36 51)(37 52)(38 53)(39 54)(40 55)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(111 149)(112 150)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 133)(122 134)(123 135)(124 136)(125 137)(126 138)(127 139)(128 140)(129 131)(130 132)
(1 112 38 92 23 130 45 110)(2 111 39 91 24 129 46 109)(3 120 40 100 25 128 47 108)(4 119 31 99 26 127 48 107)(5 118 32 98 27 126 49 106)(6 117 33 97 28 125 50 105)(7 116 34 96 29 124 41 104)(8 115 35 95 30 123 42 103)(9 114 36 94 21 122 43 102)(10 113 37 93 22 121 44 101)(11 90 146 69 158 72 138 52)(12 89 147 68 159 71 139 51)(13 88 148 67 160 80 140 60)(14 87 149 66 151 79 131 59)(15 86 150 65 152 78 132 58)(16 85 141 64 153 77 133 57)(17 84 142 63 154 76 134 56)(18 83 143 62 155 75 135 55)(19 82 144 61 156 74 136 54)(20 81 145 70 157 73 137 53)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H5A5B8A···8H8I8J8K8L10A···10F10G···10R20A···20H20I···20T
order1222222244444444558···8888810···1010···1020···2020···20
size11112244111122442210···10202020202···24···42···24···4

68 irreducible representations

dim111111122222224
type++++++++--
imageC1C2C2C2C2C4C4D4D5D10Dic5Dic5C8○D4C5⋊D4D4.Dic5
kernel(D4×C10).24C4C20.55D4C22×C52C8C2×C4.Dic5C10×C4○D4D4×C10Q8×C10C2×C20C2×C4○D4C22×C4C2×D4C2×Q8C10C2×C4C2
# reps1411162426628168

Matrix representation of (D4×C10).24C4 in GL6(𝔽41)

35350000
6400000
0040000
0004000
0000400
0000040
,
100000
010000
0032500
000900
0000400
0000040
,
100000
010000
0032500
0025900
000010
0000340
,
15310000
39260000
0038000
0003800
0000382
0000363

G:=sub<GL(6,GF(41))| [35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,5,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,25,0,0,0,0,5,9,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[15,39,0,0,0,0,31,26,0,0,0,0,0,0,38,0,0,0,0,0,0,38,0,0,0,0,0,0,38,36,0,0,0,0,2,3] >;

(D4×C10).24C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})._{24}C_4
% in TeX

G:=Group("(D4xC10).24C4");
// GroupNames label

G:=SmallGroup(320,861);
// by ID

G=gap.SmallGroup(320,861);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^2=1,d^4=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^5*b^2*c>;
// generators/relations

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