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## G = C42⋊3Dic5order 320 = 26·5

### 3rd semidirect product of C42 and Dic5 acting via Dic5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42⋊3Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — D4×C10 — C23⋊Dic5 — C42⋊3Dic5
 Lower central C5 — C10 — C2×C10 — C2×C20 — C42⋊3Dic5
 Upper central C1 — C2 — C22 — C2×D4 — C4⋊1D4

Generators and relations for C423Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 366 in 86 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C10, C10, C42, C22⋊C4, C2×D4, C2×D4, Dic5, C20, C2×C10, C2×C10, C23⋊C4, C41D4, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C42⋊C4, C23.D5, C4×C20, D4×C10, D4×C10, C23⋊Dic5, C5×C41D4, C423Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C23⋊C4, C2×Dic5, C5⋊D4, C42⋊C4, C23.D5, C23⋊Dic5, C423Dic5

Smallest permutation representation of C423Dic5
On 40 points
Generators in S40
(1 30 25 6)(2 7 26 21)(3 22 27 8)(4 9 28 23)(5 24 29 10)
(1 30 25 6)(2 7 26 21)(3 22 27 8)(4 9 28 23)(5 24 29 10)(11 16 35 40)(12 31 36 17)(13 18 37 32)(14 33 38 19)(15 20 39 34)
(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 35 6 40)(2 34 7 39)(3 33 8 38)(4 32 9 37)(5 31 10 36)(11 30 16 25)(12 29 17 24)(13 28 18 23)(14 27 19 22)(15 26 20 21)

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20L order 1 2 2 2 2 2 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 2 4 4 8 4 4 4 40 40 40 40 2 2 2 ··· 2 8 ··· 8 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + - - + + + image C1 C2 C2 C4 C4 D4 D5 Dic5 Dic5 D10 C5⋊D4 C23⋊C4 C42⋊C4 C23⋊Dic5 C42⋊3Dic5 kernel C42⋊3Dic5 C23⋊Dic5 C5×C4⋊1D4 C4×C20 D4×C10 C22×C10 C4⋊1D4 C42 C2×D4 C2×D4 C23 C10 C5 C2 C1 # reps 1 2 1 2 2 2 2 2 2 2 8 1 2 4 8

Matrix representation of C423Dic5 in GL4(𝔽41) generated by

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

C423Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_3{\rm Dic}_5
% in TeX

G:=Group("C4^2:3Dic5");
// GroupNames label

G:=SmallGroup(320,103);
// by ID

G=gap.SmallGroup(320,103);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,297,136,1684,12550]);
// Polycyclic

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

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