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G = C40.9Q8order 320 = 26·5

9th non-split extension by C40 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.9Q8, C40.90D4, C8.9Dic10, M5(2).2D5, C52C8.1C8, C4.13(C8×D5), C54(C8.C8), C20.29(C2×C8), C10.18(C4⋊C8), C20.72(C4⋊C4), (C2×C8).266D10, C8.50(C5⋊D4), (C4×Dic5).7C4, C20.4C8.7C2, (C8×Dic5).22C2, (C5×M5(2)).4C2, (C2×C40).218C22, C22.5(C8⋊D5), (C2×C10).12M4(2), C2.5(C20.8Q8), C4.28(C10.D4), (C2×C52C8).9C4, (C2×C4).135(C4×D5), (C2×C20).223(C2×C4), SmallGroup(320,69)

Series: Derived Chief Lower central Upper central

C1C20 — C40.9Q8
C1C5C10C20C40C2×C40C8×Dic5 — C40.9Q8
C5C10C20 — C40.9Q8
C1C8C2×C8M5(2)

Generators and relations for C40.9Q8
 G = < a,b,c | a40=1, b4=a30, c2=a35b2, bab-1=a21, cac-1=a9, cbc-1=a35b3 >

2C2
10C4
10C4
2C10
5C8
5C8
10C2×C4
2Dic5
2Dic5
2C16
5C42
5C2×C8
10C16
2C2×Dic5
5M5(2)
5C4×C8
2C52C16
2C80
5C8.C8

Smallest permutation representation of C40.9Q8
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 54 16 49 31 44 6 79 21 74 36 69 11 64 26 59)(2 75 17 70 32 65 7 60 22 55 37 50 12 45 27 80)(3 56 18 51 33 46 8 41 23 76 38 71 13 66 28 61)(4 77 19 72 34 67 9 62 24 57 39 52 14 47 29 42)(5 58 20 53 35 48 10 43 25 78 40 73 15 68 30 63)
(1 6 11 16 21 26 31 36)(2 15 12 25 22 35 32 5)(3 24 13 34 23 4 33 14)(7 20 17 30 27 40 37 10)(8 29 18 39 28 9 38 19)(41 72 71 62 61 52 51 42)(43 50 73 80 63 70 53 60)(44 59 74 49 64 79 54 69)(45 68 75 58 65 48 55 78)(46 77 76 67 66 57 56 47)

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

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

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

68 conjugacy classes

class 1 2A2B4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H8I8J10A10B10C10D16A16B16C16D16E16F16G16H20A20B20C20D20E20F40A···40H40I40J40K40L80A···80P
order122444444455888888888810101010161616161616161620202020202040···404040404080···80
size11211210101010221111221010101022444444202020202222442···244444···4

68 irreducible representations

dim1111111222222222224
type+++++-++-
imageC1C2C2C2C4C4C8D4Q8D5M4(2)D10Dic10C5⋊D4C4×D5C8.C8C8×D5C8⋊D5C40.9Q8
kernelC40.9Q8C20.4C8C8×Dic5C5×M5(2)C2×C52C8C4×Dic5C52C8C40C40M5(2)C2×C10C2×C8C8C8C2×C4C5C4C22C1
# reps1111228112224448888

Matrix representation of C40.9Q8 in GL4(𝔽241) generated by

2113600
03000
00052
00190190
,
21120100
1913000
001770
000177
,
3017000
023300
00190189
005051
G:=sub<GL(4,GF(241))| [211,0,0,0,36,30,0,0,0,0,0,190,0,0,52,190],[211,191,0,0,201,30,0,0,0,0,177,0,0,0,0,177],[30,0,0,0,170,233,0,0,0,0,190,50,0,0,189,51] >;

C40.9Q8 in GAP, Magma, Sage, TeX

C_{40}._9Q_8
% in TeX

G:=Group("C40.9Q8");
// GroupNames label

G:=SmallGroup(320,69);
// by ID

G=gap.SmallGroup(320,69);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,141,36,100,570,136,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=1,b^4=a^30,c^2=a^35*b^2,b*a*b^-1=a^21,c*a*c^-1=a^9,c*b*c^-1=a^35*b^3>;
// generators/relations

Export

Subgroup lattice of C40.9Q8 in TeX

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