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
| C22 — C10×A4 |
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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