metabelian, soluble, monomial, A-group
Aliases: C112⋊C4, C11⋊D11.C2, SmallGroup(484,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C112 — C112⋊C4 |
Generators and relations for C112⋊C4
G = < a,b,c | a11=b11=c4=1, ab=ba, cac-1=a3b6, cbc-1=a2b8 >
Permutation representations of C112⋊C4
►On 22 points - transitive group 22T8
Generators in S22
(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)
(1 5 9 2 6 10 3 7 11 4 8)(12 21 19 17 15 13 22 20 18 16 14)
(1 14)(2 12 11 16)(3 21 10 18)(4 19 9 20)(5 17 8 22)(6 15 7 13)
G:=sub<Sym(22)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,5,9,2,6,10,3,7,11,4,8)(12,21,19,17,15,13,22,20,18,16,14), (1,14)(2,12,11,16)(3,21,10,18)(4,19,9,20)(5,17,8,22)(6,15,7,13)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22), (1,5,9,2,6,10,3,7,11,4,8)(12,21,19,17,15,13,22,20,18,16,14), (1,14)(2,12,11,16)(3,21,10,18)(4,19,9,20)(5,17,8,22)(6,15,7,13) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22)], [(1,5,9,2,6,10,3,7,11,4,8),(12,21,19,17,15,13,22,20,18,16,14)], [(1,14),(2,12,11,16),(3,21,10,18),(4,19,9,20),(5,17,8,22),(6,15,7,13)]])
G:=TransitiveGroup(22,8);
34 conjugacy classes
| class | 1 | 2 | 4A | 4B | 11A | ··· | 11AD |
| order | 1 | 2 | 4 | 4 | 11 | ··· | 11 |
| size | 1 | 121 | 121 | 121 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 4 |
| type | + | + | + | |
| image | C1 | C2 | C4 | C112⋊C4 |
| kernel | C112⋊C4 | C11⋊D11 | C112 | C1 |
| # reps | 1 | 1 | 2 | 30 |
Matrix representation of C112⋊C4 ►in GL4(𝔽89) generated by
| 18 | 56 | 0 | 0 |
| 33 | 78 | 0 | 0 |
| 0 | 0 | 53 | 11 |
| 0 | 0 | 78 | 33 |
| 0 | 1 | 0 | 0 |
| 88 | 71 | 0 | 0 |
| 0 | 0 | 78 | 33 |
| 0 | 0 | 56 | 18 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 71 | 88 | 0 | 0 |
G:=sub<GL(4,GF(89))| [18,33,0,0,56,78,0,0,0,0,53,78,0,0,11,33],[0,88,0,0,1,71,0,0,0,0,78,56,0,0,33,18],[0,0,1,71,0,0,0,88,1,0,0,0,0,1,0,0] >;
C112⋊C4 in GAP, Magma, Sage, TeX
C_{11}^2\rtimes C_4 % in TeX
G:=Group("C11^2:C4"); // GroupNames label
G:=SmallGroup(484,8);
// by ID
G=gap.SmallGroup(484,8);
# by ID
G:=PCGroup([4,-2,-2,-11,11,8,3026,246,2691,3527]);
// Polycyclic
G:=Group<a,b,c|a^11=b^11=c^4=1,a*b=b*a,c*a*c^-1=a^3*b^6,c*b*c^-1=a^2*b^8>;
// generators/relations
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