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G = D10.18D8order 320 = 26·5

7th non-split extension by D10 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.18D8, C20.1C42, D10.9Q16, D10.18SD16, C4⋊C43F5, C4⋊F51C4, D5⋊C82C4, C4.6(C4×F5), C4⋊Dic58C4, C20.1(C4⋊C4), (C4×D5).3Q8, D5.(C2.D8), D5.(C4.Q8), (C4×D5).18D4, C4.16(C4⋊F5), D10.20(C4⋊C4), Dic5.3(C4⋊C4), C2.2(D20⋊C4), C51(C22.4Q16), C2.2(Q8⋊F5), (C2×Dic5).98D4, D5.1(D4⋊C4), C10.4(D4⋊C4), D5.1(Q8⋊C4), C10.4(Q8⋊C4), (C22×D5).142D4, D10.29(C22⋊C4), Dic5.2(C22⋊C4), C2.7(D10.3Q8), C22.33(C22⋊F5), C10.5(C2.C42), (C5×C4⋊C4)⋊3C4, (C2×C4⋊F5).1C2, (C2×D5⋊C8).1C2, (D5×C4⋊C4).14C2, (C2×C4).66(C2×F5), (C2×C20).32(C2×C4), (C4×D5).12(C2×C4), (C2×C4×D5).184C22, (C2×C10).24(C22⋊C4), SmallGroup(320,212)

Series: Derived Chief Lower central Upper central

C1C20 — D10.18D8
C1C5C10D10C22×D5C2×C4×D5C2×C4⋊F5 — D10.18D8
C5C10C20 — D10.18D8
C1C22C2×C4C4⋊C4

Generators and relations for D10.18D8
 G = < a,b,c,d | a10=b2=c8=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=dbd-1=a2b, dcd-1=a4bc-1 >

Subgroups: 498 in 114 conjugacy classes, 46 normal (42 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, C4⋊C4 [×5], C2×C8 [×4], C22×C4 [×3], Dic5 [×2], Dic5, C20 [×2], C20, F5 [×2], D10 [×6], C2×C10, C2×C4⋊C4 [×2], C22×C8, C5⋊C8 [×2], C4×D5 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5 [×4], C22×D5, C22.4Q16, C10.D4, C4⋊Dic5, C5×C4⋊C4, D5⋊C8 [×2], D5⋊C8, C4⋊F5 [×2], C4⋊F5, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C22×F5, D5×C4⋊C4, C2×D5⋊C8, C2×C4⋊F5, D10.18D8
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, D10.3Q8, D20⋊C4, Q8⋊F5, D10.18D8

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L 5 8A···8H10A10B10C20A···20F
order122222224444444···458···810101020···20
size111155552244101020···20410···104448···8

38 irreducible representations

dim1111111122222224444488
type+++++-+++-++++-
imageC1C2C2C2C4C4C4C4D4Q8D4D4D8SD16Q16F5C2×F5C4×F5C4⋊F5C22⋊F5D20⋊C4Q8⋊F5
kernelD10.18D8D5×C4⋊C4C2×D5⋊C8C2×C4⋊F5C4⋊Dic5C5×C4⋊C4D5⋊C8C4⋊F5C4×D5C4×D5C2×Dic5C22×D5D10D10D10C4⋊C4C2×C4C4C4C22C2C2
# reps1111224411112421122211

Matrix representation of D10.18D8 in GL6(𝔽41)

100000
010000
0000040
001111
0040000
0004000
,
4000000
0400000
0000040
0000400
0004000
0040000
,
15260000
15150000
00701414
00141407
002734270
00347734
,
3200000
090000
001000
000001
000100
0040404040

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],[15,15,0,0,0,0,26,15,0,0,0,0,0,0,7,14,27,34,0,0,0,14,34,7,0,0,14,0,27,7,0,0,14,7,0,34],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

D10.18D8 in GAP, Magma, Sage, TeX

D_{10}._{18}D_8
% in TeX

G:=Group("D10.18D8");
// GroupNames label

G:=SmallGroup(320,212);
// by ID

G=gap.SmallGroup(320,212);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1684,851,102,6278,3156]);
// Polycyclic

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

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