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G = C8.24D20order 320 = 26·5

10th non-split extension by C8 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.24D20, C40.42D4, D20.22D4, Dic10.22D4, M4(2).11D10, C4.137(D4×D5), C8.C47D5, C4.58(C2×D20), (C2×C8).72D10, C8⋊D10.2C2, C20.138(C2×D4), C53(D4.3D4), D20.3C49C2, C4.12D204C2, C8.D1010C2, C20.46D44C2, C10.51(C4⋊D4), C2.24(C4⋊D20), (C2×C20).314C23, (C2×C40).155C22, C4○D20.41C22, (C2×D20).93C22, C22.8(Q82D5), (C5×M4(2)).8C22, C4.Dic5.39C22, (C2×Dic10).99C22, (C5×C8.C4)⋊8C2, (C2×C40⋊C2)⋊26C2, (C2×C10).5(C4○D4), (C2×C4).115(C22×D5), SmallGroup(320,525)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C8.24D20
C1C5C10C20C2×C20C4○D20D20.3C4 — C8.24D20
C5C10C2×C20 — C8.24D20
C1C2C2×C4C8.C4

Generators and relations for C8.24D20
 G = < a,b,c | a40=1, b4=c2=a20, bab-1=a11, cac-1=a19, cbc-1=b3 >

Subgroups: 510 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, D5 [×2], C10, C10, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8, C40 [×2], C40 [×2], Dic10, Dic10 [×2], C4×D5, D20, D20 [×2], C2×Dic5, C5⋊D4, C2×C20, C22×D5, D4.3D4, C8×D5, C8⋊D5, C40⋊C2 [×4], D40, Dic20, C4.Dic5, C2×C40, C5×M4(2) [×2], C2×Dic10, C2×D20, C4○D20, C20.46D4, C4.12D20, C5×C8.C4, D20.3C4, C2×C40⋊C2, C8⋊D10, C8.D10, C8.24D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C22×D5, D4.3D4, C2×D20, D4×D5, Q82D5, C4⋊D20, C8.24D20

Smallest permutation representation of C8.24D20
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 60 31 70 21 80 11 50)(2 71 32 41 22 51 12 61)(3 42 33 52 23 62 13 72)(4 53 34 63 24 73 14 43)(5 64 35 74 25 44 15 54)(6 75 36 45 26 55 16 65)(7 46 37 56 27 66 17 76)(8 57 38 67 28 77 18 47)(9 68 39 78 29 48 19 58)(10 79 40 49 30 59 20 69)
(1 36 21 16)(2 15 22 35)(3 34 23 14)(4 13 24 33)(5 32 25 12)(6 11 26 31)(7 30 27 10)(8 9 28 29)(17 20 37 40)(18 39 38 19)(41 54 61 74)(42 73 62 53)(43 52 63 72)(44 71 64 51)(45 50 65 70)(46 69 66 49)(47 48 67 68)(55 80 75 60)(56 59 76 79)(57 78 77 58)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D5A5B8A8B8C8D8E8F8G10A10B10C10D20A20B20C20D20E20F40A···40H40I···40P
order1222244445588888881010101020202020202040···4040···40
size11220402220402222488202022442222444···48···8

44 irreducible representations

dim11111111222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D20D4.3D4D4×D5Q82D5C8.24D20
kernelC8.24D20C20.46D4C4.12D20C5×C8.C4D20.3C4C2×C40⋊C2C8⋊D10C8.D10C40Dic10D20C8.C4C2×C10C2×C8M4(2)C8C5C4C22C1
# reps11111111211222482228

Matrix representation of C8.24D20 in GL4(𝔽41) generated by

391400
273700
00272
003915
,
0010
0001
22800
133900
,
153900
312600
00414
003137
G:=sub<GL(4,GF(41))| [39,27,0,0,14,37,0,0,0,0,27,39,0,0,2,15],[0,0,2,13,0,0,28,39,1,0,0,0,0,1,0,0],[15,31,0,0,39,26,0,0,0,0,4,31,0,0,14,37] >;

C8.24D20 in GAP, Magma, Sage, TeX

C_8._{24}D_{20}
% in TeX

G:=Group("C8.24D20");
// GroupNames label

G:=SmallGroup(320,525);
// by ID

G=gap.SmallGroup(320,525);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,58,1123,136,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=1,b^4=c^2=a^20,b*a*b^-1=a^11,c*a*c^-1=a^19,c*b*c^-1=b^3>;
// generators/relations

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