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G = D10.10D8order 320 = 26·5

6th non-split extension by D10 of D8 acting via D8/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.10D8, C20.9C42, D10.5Q16, D10.9SD16, C4⋊F55C4, (C2×C8)⋊4F5, (C2×C40)⋊2C4, C4.2(C4×F5), (C4×D5).108D4, C2.3(D5.D8), C2.3(C40⋊C4), C10.5(C4.Q8), C10.7(C2.D8), C52(C22.4Q16), (C2×Dic5).31Q8, (C22×D5).96D4, C4.24(C22⋊F5), C22.19(C4⋊F5), D5.2(D4⋊C4), C20.24(C22⋊C4), Dic5.24(C4⋊C4), D5.2(Q8⋊C4), D10.31(C22⋊C4), C2.9(D10.3Q8), C10.8(C2.C42), (D5×C2×C8).12C2, (C2×C52C8)⋊16C4, (C2×C4⋊F5).8C2, (C4×D5).62(C2×C4), (C2×C4).124(C2×F5), (C2×C10).12(C4⋊C4), (C2×C20).118(C2×C4), (C2×C4×D5).387C22, SmallGroup(320,231)

Series: Derived Chief Lower central Upper central

C1C20 — D10.10D8
C1C5C10D10C4×D5C2×C4×D5C2×C4⋊F5 — D10.10D8
C5C10C20 — D10.10D8
C1C22C2×C4C2×C8

Generators and relations for D10.10D8
 G = < a,b,c,d | a10=b2=c8=1, d2=a4b, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=a5c-1 >

Subgroups: 514 in 114 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×13], C23, D5 [×4], C10 [×3], C4⋊C4 [×6], C2×C8, C2×C8 [×3], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2×C4⋊C4 [×2], C22×C8, C52C8, C40, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×8], C22×D5, C22.4Q16, C8×D5 [×2], C2×C52C8, C2×C40, C4⋊F5 [×4], C4⋊F5 [×2], C2×C4×D5, C22×F5 [×2], D5×C2×C8, C2×C4⋊F5 [×2], D10.10D8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, F5, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C2×F5, C22.4Q16, C4×F5, C4⋊F5, C22⋊F5, C40⋊C4, D5.D8, D10.3Q8, D10.10D8

Smallest permutation representation of D10.10D8
On 80 points
Generators in S80
(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)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 30)(22 29)(23 28)(24 27)(25 26)(31 34)(32 33)(35 40)(36 39)(37 38)(41 48)(42 47)(43 46)(44 45)(49 50)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)(71 74)(72 73)(75 80)(76 79)(77 78)
(1 16 50 58 21 73 33 65)(2 17 41 59 22 74 34 66)(3 18 42 60 23 75 35 67)(4 19 43 51 24 76 36 68)(5 20 44 52 25 77 37 69)(6 11 45 53 26 78 38 70)(7 12 46 54 27 79 39 61)(8 13 47 55 28 80 40 62)(9 14 48 56 29 71 31 63)(10 15 49 57 30 72 32 64)
(1 70 6 65)(2 67 5 68)(3 64 4 61)(7 62 10 63)(8 69 9 66)(11 38 16 33)(12 35 15 36)(13 32 14 39)(17 40 20 31)(18 37 19 34)(21 53 26 58)(22 60 25 51)(23 57 24 54)(27 55 30 56)(28 52 29 59)(41 75 44 76)(42 72 43 79)(45 73 50 78)(46 80 49 71)(47 77 48 74)

G:=sub<Sym(80)| (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,34)(32,33)(35,40)(36,39)(37,38)(41,48)(42,47)(43,46)(44,45)(49,50)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70)(71,74)(72,73)(75,80)(76,79)(77,78), (1,16,50,58,21,73,33,65)(2,17,41,59,22,74,34,66)(3,18,42,60,23,75,35,67)(4,19,43,51,24,76,36,68)(5,20,44,52,25,77,37,69)(6,11,45,53,26,78,38,70)(7,12,46,54,27,79,39,61)(8,13,47,55,28,80,40,62)(9,14,48,56,29,71,31,63)(10,15,49,57,30,72,32,64), (1,70,6,65)(2,67,5,68)(3,64,4,61)(7,62,10,63)(8,69,9,66)(11,38,16,33)(12,35,15,36)(13,32,14,39)(17,40,20,31)(18,37,19,34)(21,53,26,58)(22,60,25,51)(23,57,24,54)(27,55,30,56)(28,52,29,59)(41,75,44,76)(42,72,43,79)(45,73,50,78)(46,80,49,71)(47,77,48,74)>;

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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,34)(32,33)(35,40)(36,39)(37,38)(41,48)(42,47)(43,46)(44,45)(49,50)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70)(71,74)(72,73)(75,80)(76,79)(77,78), (1,16,50,58,21,73,33,65)(2,17,41,59,22,74,34,66)(3,18,42,60,23,75,35,67)(4,19,43,51,24,76,36,68)(5,20,44,52,25,77,37,69)(6,11,45,53,26,78,38,70)(7,12,46,54,27,79,39,61)(8,13,47,55,28,80,40,62)(9,14,48,56,29,71,31,63)(10,15,49,57,30,72,32,64), (1,70,6,65)(2,67,5,68)(3,64,4,61)(7,62,10,63)(8,69,9,66)(11,38,16,33)(12,35,15,36)(13,32,14,39)(17,40,20,31)(18,37,19,34)(21,53,26,58)(22,60,25,51)(23,57,24,54)(27,55,30,56)(28,52,29,59)(41,75,44,76)(42,72,43,79)(45,73,50,78)(46,80,49,71)(47,77,48,74) );

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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,30),(22,29),(23,28),(24,27),(25,26),(31,34),(32,33),(35,40),(36,39),(37,38),(41,48),(42,47),(43,46),(44,45),(49,50),(51,54),(52,53),(55,60),(56,59),(57,58),(61,68),(62,67),(63,66),(64,65),(69,70),(71,74),(72,73),(75,80),(76,79),(77,78)], [(1,16,50,58,21,73,33,65),(2,17,41,59,22,74,34,66),(3,18,42,60,23,75,35,67),(4,19,43,51,24,76,36,68),(5,20,44,52,25,77,37,69),(6,11,45,53,26,78,38,70),(7,12,46,54,27,79,39,61),(8,13,47,55,28,80,40,62),(9,14,48,56,29,71,31,63),(10,15,49,57,30,72,32,64)], [(1,70,6,65),(2,67,5,68),(3,64,4,61),(7,62,10,63),(8,69,9,66),(11,38,16,33),(12,35,15,36),(13,32,14,39),(17,40,20,31),(18,37,19,34),(21,53,26,58),(22,60,25,51),(23,57,24,54),(27,55,30,56),(28,52,29,59),(41,75,44,76),(42,72,43,79),(45,73,50,78),(46,80,49,71),(47,77,48,74)])

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4L 5 8A8B8C8D8E8F8G8H10A10B10C20A20B20C20D40A···40H
order1222222244444···45888888881010102020202040···40
size1111555522101020···20422221010101044444444···4

44 irreducible representations

dim1111112222224444444
type++++-++-+++
imageC1C2C2C4C4C4D4Q8D4D8SD16Q16F5C2×F5C4×F5C22⋊F5C4⋊F5C40⋊C4D5.D8
kernelD10.10D8D5×C2×C8C2×C4⋊F5C2×C52C8C2×C40C4⋊F5C4×D5C2×Dic5C22×D5D10D10D10C2×C8C2×C4C4C4C22C2C2
# reps1122282112421122244

Matrix representation of D10.10D8 in GL6(𝔽41)

100000
010000
0000040
001111
0040000
0004000
,
4000000
0400000
0000040
0000400
0004000
0040000
,
22220000
1520000
0032000
0003200
0000320
0000032
,
34340000
1370000
0032000
0000032
0003200
009999

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40,0,0,0],[22,15,0,0,0,0,22,2,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[34,13,0,0,0,0,34,7,0,0,0,0,0,0,32,0,0,9,0,0,0,0,32,9,0,0,0,0,0,9,0,0,0,32,0,9] >;

D10.10D8 in GAP, Magma, Sage, TeX

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

G:=Group("D10.10D8");
// GroupNames label

G:=SmallGroup(320,231);
// by ID

G=gap.SmallGroup(320,231);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,176,1123,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^4*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=a^5*c^-1>;
// generators/relations

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