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G = C102.C4  order 400 = 24·52

4th non-split extension by C102 of C4 acting faithfully

metabelian, supersoluble, monomial

Aliases: C102.4C4, C52⋊12M4(2), Dic5.3Dic5, Dic5.15D10, C52⋊3C8⋊6C2, C10.40(C2×F5), (C2×C10).11F5, C5⋊2(C4.Dic5), (C2×Dic5).5D5, (C5×Dic5).7C4, C10.6(C2×Dic5), (C2×C10).2Dic5, C22.(D5.D5), C5⋊5(C22.F5), (C10×Dic5).6C2, (C5×Dic5).19C22, C2.6(C2×D5.D5), (C5×C10).25(C2×C4), SmallGroup(400,147)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C10 — C102.C4
Chief series
C1 — C5 — C52 — C5×C10 — C5×Dic5 — C52⋊3C8 — C102.C4
Lower central
C52 — C5×C10 — C102.C4
Upper central
C1 — C2 — C22

Generators and relations for C102.C4
 G = < a,b,c | a10=b10=1, c4=b5, ab=ba, cac-1=a-1b5, cbc-1=b7 >

C1
C2
2C2
C5
C5
4C5
C22
5C4
5C4
C10
C10
2C10
2C10
4C10
4C10
4C10
C52
5C2×C4
25C8
25C8
Dic5
C2×C10
C2×C10
Dic5
4C2×C10
5C20
5C20
C5×C10
2C5×C10
25M4(2)
C2×Dic5
5C2×C20
5C5⋊C8
5C5⋊2C8
5C5⋊C8
5C5⋊2C8
C5×Dic5
C102
C5×Dic5
5C4.Dic5
5C22.F5
C52⋊3C8
C10×Dic5
C52⋊3C8
C102.C4

Smallest permutation representation of C102.C4
►On 40 points
Generators in S40
(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)

(1 6 4 9 2 7 5 10 3 8)(11 20 13 17 15 19 12 16 14 18)(21 24 27 30 23 26 29 22 25 28)(31 38 35 32 39 36 33 40 37 34)

(1 38 20 29 7 33 12 24)(2 32 16 23 8 37 13 28)(3 36 17 27 9 31 14 22)(4 40 18 21 10 35 15 26)(5 34 19 25 6 39 11 30)
 
G:=sub<Sym(40)| (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), (1,6,4,9,2,7,5,10,3,8)(11,20,13,17,15,19,12,16,14,18)(21,24,27,30,23,26,29,22,25,28)(31,38,35,32,39,36,33,40,37,34), (1,38,20,29,7,33,12,24)(2,32,16,23,8,37,13,28)(3,36,17,27,9,31,14,22)(4,40,18,21,10,35,15,26)(5,34,19,25,6,39,11,30)>;
 
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), (1,6,4,9,2,7,5,10,3,8)(11,20,13,17,15,19,12,16,14,18)(21,24,27,30,23,26,29,22,25,28)(31,38,35,32,39,36,33,40,37,34), (1,38,20,29,7,33,12,24)(2,32,16,23,8,37,13,28)(3,36,17,27,9,31,14,22)(4,40,18,21,10,35,15,26)(5,34,19,25,6,39,11,30) );
 
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)], [(1,6,4,9,2,7,5,10,3,8),(11,20,13,17,15,19,12,16,14,18),(21,24,27,30,23,26,29,22,25,28),(31,38,35,32,39,36,33,40,37,34)], [(1,38,20,29,7,33,12,24),(2,32,16,23,8,37,13,28),(3,36,17,27,9,31,14,22),(4,40,18,21,10,35,15,26),(5,34,19,25,6,39,11,30)]])
 

46 conjugacy classes

class 1 2A2B4A4B4C5A5B5C···5G8A8B8C8D10A···10F10G···10U20A···20H
order122444555···5888810···1010···1020···20
size1125510224···4505050502···24···410···10

46 irreducible representations

dim11111222222444444
type++++-+-++-
imageC1C2C2C4C4D5M4(2)Dic5D10Dic5C4.Dic5F5C2×F5C22.F5D5.D5C2×D5.D5C102.C4
kernelC102.C4C52⋊3C8C10×Dic5C5×Dic5C102C2×Dic5C52Dic5Dic5C2×C10C5C2×C10C10C5C22C2C1
# reps12122222228112448

Matrix representation of C102.C4 ►in GL6(𝔽41)

1600000
0230000
0018000
0001800
0000160
0000016
,
4000000
0400000
0016000
0001800
0000370
0000010
,
040000
800000
000010
000001
000100
001000

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,23,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,0,0,0,0,0,0,18,0,0,0,0,0,0,37,0,0,0,0,0,0,10],[0,8,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C102.C4 in GAP, Magma, Sage, TeX 

C_{10}^2.C_4
 
% in TeX
 
G:=Group("C10^2.C4");
 
// GroupNames label
 
G:=SmallGroup(400,147);
 
// by ID
 
G=gap.SmallGroup(400,147);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,1924,8645,2897]);
 
// Polycyclic
 
G:=Group<a,b,c|a^10=b^10=1,c^4=b^5,a*b=b*a,c*a*c^-1=a^-1*b^5,c*b*c^-1=b^7>;
 
// generators/relations
 

Export

Subgroup lattice of C102.C4 in TeX

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