direct product, metabelian, soluble, monomial, A-group
Aliases: C11×A4, C22⋊C33, (C2×C22)⋊C3, SmallGroup(132,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C11×A4 |
Generators and relations for C11×A4
G = < a,b,c,d | a11=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C11×A4
►On 44 points
Generators in S44
(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 31 32 33)(34 35 36 37 38 39 40 41 42 43 44)
(1 32)(2 33)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 34)(20 35)(21 36)(22 37)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 30)(13 31)(14 32)(15 33)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(22 29)
(12 30 38)(13 31 39)(14 32 40)(15 33 41)(16 23 42)(17 24 43)(18 25 44)(19 26 34)(20 27 35)(21 28 36)(22 29 37)
G:=sub<Sym(44)| (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44), (1,32)(2,33)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,34)(20,35)(21,36)(22,37), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,30)(13,31)(14,32)(15,33)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(22,29), (12,30,38)(13,31,39)(14,32,40)(15,33,41)(16,23,42)(17,24,43)(18,25,44)(19,26,34)(20,27,35)(21,28,36)(22,29,37)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38,39,40,41,42,43,44), (1,32)(2,33)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,34)(20,35)(21,36)(22,37), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,30)(13,31)(14,32)(15,33)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(22,29), (12,30,38)(13,31,39)(14,32,40)(15,33,41)(16,23,42)(17,24,43)(18,25,44)(19,26,34)(20,27,35)(21,28,36)(22,29,37) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33),(34,35,36,37,38,39,40,41,42,43,44)], [(1,32),(2,33),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,34),(20,35),(21,36),(22,37)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,30),(13,31),(14,32),(15,33),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(22,29)], [(12,30,38),(13,31,39),(14,32,40),(15,33,41),(16,23,42),(17,24,43),(18,25,44),(19,26,34),(20,27,35),(21,28,36),(22,29,37)]])
C11×A4 is a maximal subgroup of
C11⋊S4
44 conjugacy classes
| class | 1 | 2 | 3A | 3B | 11A | ··· | 11J | 22A | ··· | 22J | 33A | ··· | 33T |
| order | 1 | 2 | 3 | 3 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 |
| size | 1 | 3 | 4 | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C3 | C11 | C33 | A4 | C11×A4 |
| kernel | C11×A4 | C2×C22 | A4 | C22 | C11 | C1 |
| # reps | 1 | 2 | 10 | 20 | 1 | 10 |
Matrix representation of C11×A4 ►in GL3(𝔽67) generated by
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 0 | 1 | 66 |
| 1 | 0 | 66 |
| 0 | 0 | 66 |
| 66 | 0 | 0 |
| 66 | 0 | 1 |
| 66 | 1 | 0 |
| 1 | 66 | 0 |
| 0 | 66 | 1 |
| 0 | 66 | 0 |
G:=sub<GL(3,GF(67))| [9,0,0,0,9,0,0,0,9],[0,1,0,1,0,0,66,66,66],[66,66,66,0,0,1,0,1,0],[1,0,0,66,66,66,0,1,0] >;
C11×A4 in GAP, Magma, Sage, TeX
C_{11}\times A_4 % in TeX
G:=Group("C11xA4"); // GroupNames label
G:=SmallGroup(132,6);
// by ID
G=gap.SmallGroup(132,6);
# by ID
G:=PCGroup([4,-3,-11,-2,2,794,1587]);
// Polycyclic
G:=Group<a,b,c,d|a^11=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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