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## G = C11×A4order 132 = 22·3·11

### Direct product of C11 and A4

Aliases: C11×A4, C22⋊C33, (C2×C22)⋊C3, SmallGroup(132,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C11×A4
 Chief series C1 — C22 — C2×C22 — C11×A4
 Lower central C22 — C11×A4
 Upper central C1 — C11

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 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 23)(12 40)(13 41)(14 42)(15 43)(16 44)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 23)
(12 24 40)(13 25 41)(14 26 42)(15 27 43)(16 28 44)(17 29 34)(18 30 35)(19 31 36)(20 32 37)(21 33 38)(22 23 39)

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), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,23)(12,40)(13,41)(14,42)(15,43)(16,44)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,23), (12,24,40)(13,25,41)(14,26,42)(15,27,43)(16,28,44)(17,29,34)(18,30,35)(19,31,36)(20,32,37)(21,33,38)(22,23,39)>;

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), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,23)(12,40)(13,41)(14,42)(15,43)(16,44)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39), (1,40)(2,41)(3,42)(4,43)(5,44)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,23), (12,24,40)(13,25,41)(14,26,42)(15,27,43)(16,28,44)(17,29,34)(18,30,35)(19,31,36)(20,32,37)(21,33,38)(22,23,39) );

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)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,23),(12,40),(13,41),(14,42),(15,43),(16,44),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,23)], [(12,24,40),(13,25,41),(14,26,42),(15,27,43),(16,28,44),(17,29,34),(18,30,35),(19,31,36),(20,32,37),(21,33,38),(22,23,39)])

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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