metabelian, soluble, monomial, A-group
Aliases: C52⋊A4, C102⋊3C3, C22⋊(C52⋊C3), SmallGroup(300,43)
Series: Derived ►Chief ►Lower central ►Upper central
| C102 — C52⋊A4 |
Generators and relations for C52⋊A4
G = < a,b,c,d,e | a5=b5=c2=d2=e3=1, ebe-1=ab=ba, ac=ca, ad=da, eae-1=a3b2, bc=cb, bd=db, ece-1=cd=dc, ede-1=c >
Permutation representations of C52⋊A4
►On 30 points - transitive group 30T70
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 5 4 3 2)(6 10 9 8 7)(11 13 15 12 14)(16 18 20 17 19)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 21 12)(2 22 14)(3 23 11)(4 24 13)(5 25 15)(6 26 17)(7 27 19)(8 28 16)(9 29 18)(10 30 20)
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), (1,5,4,3,2)(6,10,9,8,7)(11,13,15,12,14)(16,18,20,17,19), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,21,12)(2,22,14)(3,23,11)(4,24,13)(5,25,15)(6,26,17)(7,27,19)(8,28,16)(9,29,18)(10,30,20)>;
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), (1,5,4,3,2)(6,10,9,8,7)(11,13,15,12,14)(16,18,20,17,19), (11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20), (1,21,12)(2,22,14)(3,23,11)(4,24,13)(5,25,15)(6,26,17)(7,27,19)(8,28,16)(9,29,18)(10,30,20) );
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)], [(1,5,4,3,2),(6,10,9,8,7),(11,13,15,12,14),(16,18,20,17,19)], [(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,21,12),(2,22,14),(3,23,11),(4,24,13),(5,25,15),(6,26,17),(7,27,19),(8,28,16),(9,29,18),(10,30,20)]])
G:=TransitiveGroup(30,70);
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | ··· | 5H | 10A | ··· | 10X |
| order | 1 | 2 | 3 | 3 | 5 | ··· | 5 | 10 | ··· | 10 |
| size | 1 | 3 | 100 | 100 | 3 | ··· | 3 | 3 | ··· | 3 |
36 irreducible representations
| dim | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||
| image | C1 | C3 | A4 | C52⋊C3 | C52⋊A4 |
| kernel | C52⋊A4 | C102 | C52 | C22 | C1 |
| # reps | 1 | 2 | 1 | 8 | 24 |
Matrix representation of C52⋊A4 ►in GL3(𝔽11) generated by
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 3 |
| 5 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 9 |
| 10 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 10 |
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(11))| [9,0,0,0,9,0,0,0,3],[5,0,0,0,1,0,0,0,9],[10,0,0,0,1,0,0,0,10],[10,0,0,0,10,0,0,0,1],[0,1,0,0,0,1,1,0,0] >;
C52⋊A4 in GAP, Magma, Sage, TeX
C_5^2\rtimes A_4
% in TeX
G:=Group("C5^2:A4"); // GroupNames label
G:=SmallGroup(300,43);
// by ID
G=gap.SmallGroup(300,43);
# by ID
G:=PCGroup([5,-3,-2,2,-5,5,61,137,3843,5704]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^5=c^2=d^2=e^3=1,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^3*b^2,b*c=c*b,b*d=d*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
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