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## G = C10×A4order 120 = 23·3·5

### Direct product of C10 and A4

Aliases: C10×A4, C23⋊C15, C22⋊C30, (C22×C10)⋊C3, (C2×C10)⋊2C6, SmallGroup(120,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C10×A4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4
 Lower central C22 — C10×A4
 Upper central C1 — C10

Generators and relations for C10×A4
G = < a,b,c,d | a10=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Permutation representations of C10×A4
On 30 points - transitive group 30T18
Generators in S30
(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)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 23 13)(2 24 14)(3 25 15)(4 26 16)(5 27 17)(6 28 18)(7 29 19)(8 30 20)(9 21 11)(10 22 12)

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

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), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,6)(2,7)(3,8)(4,9)(5,10)(21,26)(22,27)(23,28)(24,29)(25,30), (1,23,13)(2,24,14)(3,25,15)(4,26,16)(5,27,17)(6,28,18)(7,29,19)(8,30,20)(9,21,11)(10,22,12) );

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)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,6),(2,7),(3,8),(4,9),(5,10),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,23,13),(2,24,14),(3,25,15),(4,26,16),(5,27,17),(6,28,18),(7,29,19),(8,30,20),(9,21,11),(10,22,12)])

G:=TransitiveGroup(30,18);

C10×A4 is a maximal subgroup of   A4⋊Dic5

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 10E ··· 10L 15A ··· 15H 30A ··· 30H order 1 2 2 2 3 3 5 5 5 5 6 6 10 10 10 10 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 3 3 4 4 1 1 1 1 4 4 1 1 1 1 3 ··· 3 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 A4 C2×A4 C5×A4 C10×A4 kernel C10×A4 C5×A4 C22×C10 C2×A4 C2×C10 A4 C23 C22 C10 C5 C2 C1 # reps 1 1 2 4 2 4 8 8 1 1 4 4

Matrix representation of C10×A4 in GL3(𝔽11) generated by

 6 0 0 0 6 0 0 0 6
,
 0 7 4 8 0 10 0 0 10
,
 0 4 7 0 10 0 8 10 0
,
 1 0 4 0 0 10 0 1 10
G:=sub<GL(3,GF(11))| [6,0,0,0,6,0,0,0,6],[0,8,0,7,0,0,4,10,10],[0,0,8,4,10,10,7,0,0],[1,0,0,0,0,1,4,10,10] >;

C10×A4 in GAP, Magma, Sage, TeX

C_{10}\times A_4
% in TeX

G:=Group("C10xA4");
// GroupNames label

G:=SmallGroup(120,43);
// by ID

G=gap.SmallGroup(120,43);
# by ID

G:=PCGroup([5,-2,-3,-5,-2,2,608,1134]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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