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G = C20.47D4order 160 = 25·5

4th non-split extension by C20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.47D4, C4.12D20, M4(2).2D5, (C2×C4).2D10, C22.5(C4×D5), C52(C4.10D4), C4.22(C5⋊D4), (C2×Dic5).1C4, C4.Dic5.3C2, (C2×C20).14C22, (C2×Dic10).7C2, (C5×M4(2)).2C2, C10.20(C22⋊C4), C2.10(D10⋊C4), (C2×C10).23(C2×C4), SmallGroup(160,31)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.47D4
C1C5C10C20C2×C20C2×Dic10 — C20.47D4
C5C10C2×C10 — C20.47D4
C1C2C2×C4M4(2)

Generators and relations for C20.47D4
 G = < a,b,c | a20=1, b4=c2=a10, bab-1=cac-1=a-1, cbc-1=a5b3 >

2C2
10C4
10C4
2C10
2C8
5C2×C4
5C2×C4
10C8
10Q8
10Q8
2Dic5
2Dic5
5M4(2)
5C2×Q8
2Dic10
2Dic10
2C52C8
2C40
5C4.10D4

Smallest permutation representation of C20.47D4
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 62 6 77 11 72 16 67)(2 61 7 76 12 71 17 66)(3 80 8 75 13 70 18 65)(4 79 9 74 14 69 19 64)(5 78 10 73 15 68 20 63)(21 55 36 60 31 45 26 50)(22 54 37 59 32 44 27 49)(23 53 38 58 33 43 28 48)(24 52 39 57 34 42 29 47)(25 51 40 56 35 41 30 46)
(1 50 11 60)(2 49 12 59)(3 48 13 58)(4 47 14 57)(5 46 15 56)(6 45 16 55)(7 44 17 54)(8 43 18 53)(9 42 19 52)(10 41 20 51)(21 62 31 72)(22 61 32 71)(23 80 33 70)(24 79 34 69)(25 78 35 68)(26 77 36 67)(27 76 37 66)(28 75 38 65)(29 74 39 64)(30 73 40 63)

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

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

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

31 conjugacy classes

class 1 2A2B4A4B4C4D5A5B8A8B8C8D10A10B10C10D20A20B20C20D20E20F40A···40H
order12244445588881010101020202020202040···40
size1122220202244202022442222444···4

31 irreducible representations

dim1111122222244
type++++++++--
imageC1C2C2C2C4D4D5D10D20C5⋊D4C4×D5C4.10D4C20.47D4
kernelC20.47D4C4.Dic5C5×M4(2)C2×Dic10C2×Dic5C20M4(2)C2×C4C4C4C22C5C1
# reps1111422244414

Matrix representation of C20.47D4 in GL4(𝔽41) generated by

253900
21300
002539
00213
,
001437
003927
21500
273900
,
143700
392700
001437
003927
G:=sub<GL(4,GF(41))| [25,2,0,0,39,13,0,0,0,0,25,2,0,0,39,13],[0,0,2,27,0,0,15,39,14,39,0,0,37,27,0,0],[14,39,0,0,37,27,0,0,0,0,14,39,0,0,37,27] >;

C20.47D4 in GAP, Magma, Sage, TeX

C_{20}._{47}D_4
% in TeX

G:=Group("C20.47D4");
// GroupNames label

G:=SmallGroup(160,31);
// by ID

G=gap.SmallGroup(160,31);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,121,31,362,86,297,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=1,b^4=c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^3>;
// generators/relations

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