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G = D813D10order 320 = 26·5

2nd semidirect product of D8 and D10 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D813D10, D20.28D4, C20.4C24, D4016C22, C40.40C23, D20.2C23, Dic10.28D4, Dic2014C22, Dic10.2C23, (D5×D8)⋊6C2, (C2×C8)⋊9D10, C52(D4○D8), (C10×D8)⋊3C2, (C2×D8)⋊12D5, C4.75(D4×D5), C5⋊D4.8D4, (C2×D4)⋊14D10, D83D56C2, D8⋊D55C2, (C2×C40)⋊3C22, D4⋊D51C22, C20.79(C2×D4), (C8×D5)⋊7C22, (D4×D5)⋊1C22, D407C23C2, D46D105C2, C4.4(C23×D5), D10.49(C2×D4), C4○D203C22, (C5×D8)⋊11C22, C52C8.1C23, D4.D51C22, (C5×D4).2C23, D4.2(C22×D5), (C4×D5).2C23, C22.20(D4×D5), C8.10(C22×D5), D20.3C42C2, D4.D107C2, D42D51C22, (D4×C10)⋊20C22, C40⋊C214C22, C8⋊D513C22, Dic5.55(C2×D4), (C2×C20).521C23, C10.105(C22×D4), C4.Dic528C22, C2.78(C2×D4×D5), (C2×C10).394(C2×D4), (C2×C4).229(C22×D5), SmallGroup(320,1429)

Series: Derived Chief Lower central Upper central

C1C20 — D813D10
C1C5C10C20C4×D5C4○D20D46D10 — D813D10
C5C10C20 — D813D10
C1C2C2×C4C2×D8

Generators and relations for D813D10
 G = < a,b,c,d | a8=b2=c10=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a6b, dbd=a4b, dcd=c-1 >

Subgroups: 1142 in 268 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×14], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4 [×4], D4 [×17], Q8 [×3], C23 [×6], D5 [×4], C10, C10 [×5], C2×C8, C2×C8 [×2], M4(2) [×3], D8 [×4], D8 [×5], SD16 [×6], Q16, C2×D4 [×2], C2×D4 [×10], C4○D4 [×9], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×6], C8○D4, C2×D8, C2×D8 [×2], C4○D8 [×3], C8⋊C22 [×6], 2+ 1+4 [×2], C52C8 [×2], C40 [×2], Dic10, Dic10 [×2], C4×D5 [×2], C4×D5 [×2], D20, D20 [×2], C2×Dic5 [×4], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×D5 [×4], C22×C10 [×2], D4○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C4.Dic5, D4⋊D5 [×4], D4.D5 [×4], C2×C40, C5×D8 [×4], C4○D20, C4○D20 [×2], D4×D5 [×4], D4×D5 [×2], D42D5 [×4], D42D5 [×2], C2×C5⋊D4 [×4], D4×C10 [×2], D20.3C4, D407C2, D5×D8 [×2], D8⋊D5 [×4], D83D5 [×2], D4.D10 [×2], C10×D8, D46D10 [×2], D813D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D813D10

Smallest permutation representation of D813D10
On 80 points
Generators in S80
(1 34 28 20 73 53 66 42)(2 43 67 54 74 11 29 35)(3 36 30 12 75 55 68 44)(4 45 69 56 76 13 21 37)(5 38 22 14 77 57 70 46)(6 47 61 58 78 15 23 39)(7 40 24 16 79 59 62 48)(8 49 63 60 80 17 25 31)(9 32 26 18 71 51 64 50)(10 41 65 52 72 19 27 33)
(2 29)(4 21)(6 23)(8 25)(10 27)(11 43)(12 55)(13 45)(14 57)(15 47)(16 59)(17 49)(18 51)(19 41)(20 53)(22 70)(24 62)(26 64)(28 66)(30 68)(32 50)(34 42)(36 44)(38 46)(40 48)(61 78)(63 80)(65 72)(67 74)(69 76)
(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 41)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 71)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 60)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 70)(40 69)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F5A5B8A8B8C8D8E10A···10F10G···10N20A20B20C20D40A···40H
order12222222222444444558888810···1010···102020202040···40
size11244441010202022101020202222420202···28···844444···4

50 irreducible representations

dim11111111122222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D4○D8D4×D5D4×D5D813D10
kernelD813D10D20.3C4D407C2D5×D8D8⋊D5D83D5D4.D10C10×D8D46D10Dic10D20C5⋊D4C2×D8C2×C8D8C2×D4C5C4C22C1
# reps11124221211222842228

Matrix representation of D813D10 in GL6(𝔽41)

4000000
0400000
00242400
0029000
000121229
0029291212
,
4000000
0400000
001000
00404000
000010
00400040
,
35350000
6400000
00001717
00290240
000122929
0029291212
,
35350000
4060000
00400039
0010401
004040040
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,24,29,0,29,0,0,24,0,12,29,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,40,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,29,0,29,0,0,0,0,12,29,0,0,17,24,29,12,0,0,17,0,29,12],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,1,40,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,39,1,40,1] >;

D813D10 in GAP, Magma, Sage, TeX

D_8\rtimes_{13}D_{10}
% in TeX

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

G:=SmallGroup(320,1429);
// by ID

G=gap.SmallGroup(320,1429);
# by ID

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

G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^6*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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