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G = C20.23C42  order 320 = 26·5

16th non-split extension by C20 of C42 acting via C42/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.23C42, C20.28M4(2), C5⋊C16⋊4C4, (C2×C8).1F5, (C2×C40).6C4, C5⋊4(C16⋊C4), C4.28(C4×F5), C40⋊8C4.5C2, (C4×Dic5).6C4, C10.7(C8⋊C4), C20.C8.4C2, (C2×C10).4M4(2), C22.3(C4.F5), C4.6(C22.F5), C2.4(C10.C42), C5⋊2C8.19(C2×C4), (C2×C4).122(C2×F5), (C2×C20).139(C2×C4), (C2×C5⋊2C8).212C22, SmallGroup(320,228)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — C20.23C42
Chief series
C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C20.C8 — C20.23C42
Lower central
C5 — C20 — C20.23C42
Upper central
C1 — C4 — C2×C8

Generators and relations for C20.23C42
 G = < a,b,c | a20=1, b4=a5, c4=a10, bab-1=a13, ac=ca, cbc-1=a5b >

C1
C2
2C2
C5
C4
C22
C4
20C4
C10
2C10
C2×C4
2C8
5C8
5C8
10C2×C4
C20
C2×C10
C20
4Dic5
C2×C8
5C42
5C16
5C16
5C2×C8
5C16
5C16
C2×C20
C5⋊2C8
C5⋊2C8
2C40
2C2×Dic5
5M5(2)
5C8⋊C4
5M5(2)
C4×Dic5
C5⋊C16
C5⋊C16
C2×C5⋊2C8
C2×C40
C5⋊C16
C5⋊C16
5C16⋊C4
C40⋊8C4
C20.C8
C20.C8
C20.23C42

Smallest permutation representation of C20.23C42
►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 68 24 46 6 73 29 51 11 78 34 56 16 63 39 41)(2 65 33 59 7 70 38 44 12 75 23 49 17 80 28 54)(3 62 22 52 8 67 27 57 13 72 32 42 18 77 37 47)(4 79 31 45 9 64 36 50 14 69 21 55 19 74 26 60)(5 76 40 58 10 61 25 43 15 66 30 48 20 71 35 53)

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

38 conjugacy classes

class 1 2A2B4A4B4C4D4E 5 8A8B8C8D8E8F10A10B10C16A···16H20A20B20C20D40A···40H
order12244444588888810101016···162020202040···40
size11211220204441010101044420···2044444···4

38 irreducible representations

dim111111224444444
type+++++-
imageC1C2C2C4C4C4M4(2)M4(2)F5C2×F5C16⋊C4C4×F5C22.F5C4.F5C20.23C42
kernelC20.23C42C40⋊8C4C20.C8C5⋊C16C4×Dic5C2×C40C20C2×C10C2×C8C2×C4C5C4C4C22C1
# reps112822221122228

Matrix representation of C20.23C42 ►in GL8(𝔽241)

640000000
064000000
006400000
000640000
00000001
0000240001
0000024001
0000002401
,
00100000
00010000
01000000
640000000
00001412014190
000041110231231
00001311010131
0000151151100221
,
064000000
10000000
0002400000
006400000
000017207034
0000022420734
0000342072240
000034020717

G:=sub<GL(8,GF(241))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1,1,1],[0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,141,41,131,151,0,0,0,0,20,110,10,151,0,0,0,0,141,231,10,100,0,0,0,0,90,231,131,221],[0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,17,0,34,34,0,0,0,0,207,224,207,0,0,0,0,0,0,207,224,207,0,0,0,0,34,34,0,17] >;
 

C20.23C42 in GAP, Magma, Sage, TeX 

C_{20}._{23}C_4^2
 
% in TeX
 
G:=Group("C20.23C4^2");
 
// GroupNames label
 
G:=SmallGroup(320,228);
 
// by ID
 
G=gap.SmallGroup(320,228);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,100,1123,136,102,6278,3156]);
 
// Polycyclic
 
G:=Group<a,b,c|a^20=1,b^4=a^5,c^4=a^10,b*a*b^-1=a^13,a*c=c*a,c*b*c^-1=a^5*b>;
 
// generators/relations
 

Export

Subgroup lattice of C20.23C42 in TeX

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