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G = D20.6D4order 320 = 26·5

6th non-split extension by D20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.6D4, Dic10.6D4, M4(2).5D10, C20.99(C2×D4), C4.152(D4×D5), C52C8.42D4, (C2×Q8).7D10, C8⋊D10.1C2, C4.10D43D5, C52(D4.3D4), D20.2C47C2, C20.46D48C2, C20.C231C2, C20.53D43C2, (C2×C20).11C23, C4○D20.7C22, (Q8×C10).9C22, C10.11(C4⋊D4), (C2×D20).46C22, C2.14(D10⋊D4), C4.Dic5.6C22, C22.15(C4○D20), (C5×M4(2)).14C22, (C2×Q8⋊D5)⋊1C2, (C5×C4.10D4)⋊1C2, (C2×C52C8).3C22, (C2×C4).11(C22×D5), (C2×C10).32(C4○D4), SmallGroup(320,381)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.6D4
C1C5C10C20C2×C20C4○D20D20.2C4 — D20.6D4
C5C10C2×C20 — D20.6D4
C1C2C2×C4C4.10D4

Generators and relations for D20.6D4
 G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, dcd-1=a15c3 >

Subgroups: 478 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×5], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, D5 [×2], C10, C10, C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20 [×2], C20, D10 [×3], C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8 [×2], C52C8, C40 [×2], Dic10, C4×D5, D20, D20 [×2], C5⋊D4, C2×C20, C2×C20, C5×Q8 [×2], C22×D5, D4.3D4, C8×D5, C8⋊D5, C40⋊C2, D40, C2×C52C8, C4.Dic5, Q8⋊D5 [×3], C5⋊Q16, C5×M4(2) [×2], C2×D20, C4○D20, Q8×C10, C20.53D4, C20.46D4, C5×C4.10D4, D20.2C4, C8⋊D10, C2×Q8⋊D5, C20.C23, D20.6D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C22×D5, D4.3D4, C4○D20, D4×D5 [×2], D10⋊D4, D20.6D4

Smallest permutation representation of D20.6D4
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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 80)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 70)(21 45)(22 44)(23 43)(24 42)(25 41)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)
(1 58 6 53 11 48 16 43)(2 49 7 44 12 59 17 54)(3 60 8 55 13 50 18 45)(4 51 9 46 14 41 19 56)(5 42 10 57 15 52 20 47)(21 77 26 72 31 67 36 62)(22 68 27 63 32 78 37 73)(23 79 28 74 33 69 38 64)(24 70 29 65 34 80 39 75)(25 61 30 76 35 71 40 66)
(1 65 16 80 11 75 6 70)(2 66 17 61 12 76 7 71)(3 67 18 62 13 77 8 72)(4 68 19 63 14 78 9 73)(5 69 20 64 15 79 10 74)(21 60 36 55 31 50 26 45)(22 41 37 56 32 51 27 46)(23 42 38 57 33 52 28 47)(24 43 39 58 34 53 29 48)(25 44 40 59 35 54 30 49)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A10B10C10D20A20B20C20D20E20F20G20H40A···40H
order12222444455888888810101010202020202020202040···40
size11220402282022448101020402244444488888···8

38 irreducible representations

dim1111111122222222448
type++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10C4○D20D4.3D4D4×D5D20.6D4
kernelD20.6D4C20.53D4C20.46D4C5×C4.10D4D20.2C4C8⋊D10C2×Q8⋊D5C20.C23C52C8Dic10D20C4.10D4C2×C10M4(2)C2×Q8C22C5C4C1
# reps1111111121122428242

Matrix representation of D20.6D4 in GL6(𝔽41)

4010000
3370000
000100
0040000
0004040
0037010
,
4000000
3310000
0012292417
0015152424
006402612
001362629
,
100000
010000
00537180
00375018
0000364
0090436
,
100000
010000
0015152424
0012292417
0025352629
0035282612

G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,0,40,0,37,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,12,15,6,13,0,0,29,15,40,6,0,0,24,24,26,26,0,0,17,24,12,29],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,37,0,9,0,0,37,5,0,0,0,0,18,0,36,4,0,0,0,18,4,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,12,25,35,0,0,15,29,35,28,0,0,24,24,26,26,0,0,24,17,29,12] >;

D20.6D4 in GAP, Magma, Sage, TeX

D_{20}._6D_4
% in TeX

G:=Group("D20.6D4");
// GroupNames label

G:=SmallGroup(320,381);
// by ID

G=gap.SmallGroup(320,381);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,184,297,136,1684,851,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^3>;
// generators/relations

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