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G = C10×A4order 120 = 23·3·5

Direct product of C10 and A4

direct product, metabelian, soluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C22 — C10×A4
C1C22C2×C10C5×A4 — C10×A4
C22 — C10×A4
C1C10

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 >

3C2
3C2
4C3
3C22
3C22
4C6
3C10
3C10
4C15
3C2×C10
3C2×C10
4C30

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 2A2B2C3A3B5A5B5C5D6A6B10A10B10C10D10E···10L15A···15H30A···30H
order1222335555661010101010···1015···1530···30
size11334411114411113···34···44···4

40 irreducible representations

dim111111113333
type++++
imageC1C2C3C5C6C10C15C30A4C2×A4C5×A4C10×A4
kernelC10×A4C5×A4C22×C10C2×A4C2×C10A4C23C22C10C5C2C1
# reps112424881144

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

600
060
006
,
074
8010
0010
,
047
0100
8100
,
104
0010
0110
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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Subgroup lattice of C10×A4 in TeX

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