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G = D10.13D8order 320 = 26·5

2nd non-split extension by D10 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.13D8, C2.D82D5, C2.13(D5×D8), C4⋊C4.47D10, C10.29(C2×D8), (C2×C8).27D10, C4⋊D20.8C2, D101C825C2, D205C426C2, D206C420C2, C4.80(C4○D20), C20.37(C4○D4), C10.D821C2, (C2×Dic5).56D4, C22.228(D4×D5), C53(C22.D8), (C2×C20).298C23, (C2×C40).241C22, C4.28(Q82D5), (C2×D20).85C22, (C22×D5).122D4, C2.22(Q16⋊D5), C10.70(C8.C22), C4⋊Dic5.124C22, C2.15(D10.13D4), C10.45(C22.D4), (D5×C4⋊C4)⋊7C2, (C5×C2.D8)⋊11C2, (C2×C4×D5).42C22, (C2×C10).303(C2×D4), (C5×C4⋊C4).91C22, (C2×C52C8).69C22, (C2×C4).401(C22×D5), SmallGroup(320,509)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D10.13D8
C1C5C10C2×C10C2×C20C2×C4×D5D5×C4⋊C4 — D10.13D8
C5C10C2×C20 — D10.13D8
C1C22C2×C4C2.D8

Generators and relations for D10.13D8
 G = < a,b,c,d | a10=b2=c8=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 574 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], D5 [×3], C10 [×3], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, C22×C4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, C22⋊C8, D4⋊C4 [×2], C2.D8, C2.D8, C2×C4⋊C4, C4⋊D4, C52C8, C40, C4×D5 [×4], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C22.D8, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C10.D8, D206C4, D101C8, D205C4, C5×C2.D8, D5×C4⋊C4, C4⋊D20, D10.13D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×D8, C8.C22, C22×D5, C22.D8, C4○D20, D4×D5, Q82D5, D10.13D4, D5×D8, Q16⋊D5, D10.13D8

Smallest permutation representation of D10.13D8
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)
(1 152)(2 151)(3 160)(4 159)(5 158)(6 157)(7 156)(8 155)(9 154)(10 153)(11 150)(12 149)(13 148)(14 147)(15 146)(16 145)(17 144)(18 143)(19 142)(20 141)(21 44)(22 43)(23 42)(24 41)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 131)(32 140)(33 139)(34 138)(35 137)(36 136)(37 135)(38 134)(39 133)(40 132)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 120)(58 119)(59 118)(60 117)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 110)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 100)(78 99)(79 98)(80 97)(81 123)(82 122)(83 121)(84 130)(85 129)(86 128)(87 127)(88 126)(89 125)(90 124)
(1 68 14 73 33 90 50 60)(2 69 15 74 34 81 41 51)(3 70 16 75 35 82 42 52)(4 61 17 76 36 83 43 53)(5 62 18 77 37 84 44 54)(6 63 19 78 38 85 45 55)(7 64 20 79 39 86 46 56)(8 65 11 80 40 87 47 57)(9 66 12 71 31 88 48 58)(10 67 13 72 32 89 49 59)(21 118 158 103 143 95 135 125)(22 119 159 104 144 96 136 126)(23 120 160 105 145 97 137 127)(24 111 151 106 146 98 138 128)(25 112 152 107 147 99 139 129)(26 113 153 108 148 100 140 130)(27 114 154 109 149 91 131 121)(28 115 155 110 150 92 132 122)(29 116 156 101 141 93 133 123)(30 117 157 102 142 94 134 124)
(1 143 6 148)(2 144 7 149)(3 145 8 150)(4 146 9 141)(5 147 10 142)(11 155 16 160)(12 156 17 151)(13 157 18 152)(14 158 19 153)(15 159 20 154)(21 38 26 33)(22 39 27 34)(23 40 28 35)(24 31 29 36)(25 32 30 37)(41 136 46 131)(42 137 47 132)(43 138 48 133)(44 139 49 134)(45 140 50 135)(51 96 56 91)(52 97 57 92)(53 98 58 93)(54 99 59 94)(55 100 60 95)(61 106 66 101)(62 107 67 102)(63 108 68 103)(64 109 69 104)(65 110 70 105)(71 116 76 111)(72 117 77 112)(73 118 78 113)(74 119 79 114)(75 120 80 115)(81 126 86 121)(82 127 87 122)(83 128 88 123)(84 129 89 124)(85 130 90 125)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444455888810···102020202020···2040···40
size111110104022448202020224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type++++++++++++++-+++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D8D10D10C4○D20C8.C22Q82D5D4×D5D5×D8Q16⋊D5
kernelD10.13D8C10.D8D206C4D101C8D205C4C5×C2.D8D5×C4⋊C4C4⋊D20C2×Dic5C22×D5C2.D8C20D10C4⋊C4C2×C8C4C10C4C22C2C2
# reps111111111124442812244

Matrix representation of D10.13D8 in GL6(𝔽41)

4000000
0400000
00353500
0064000
000010
000001
,
090000
3200000
00353500
0040600
0000400
0000040
,
010000
100000
0040000
0004000
000007
00003524
,
0320000
3200000
001000
000100
0000404
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,0,0,0,0,7,24],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,4,1] >;

D10.13D8 in GAP, Magma, Sage, TeX

D_{10}._{13}D_8
% in TeX

G:=Group("D10.13D8");
// GroupNames label

G:=SmallGroup(320,509);
// by ID

G=gap.SmallGroup(320,509);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,254,219,268,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
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