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## G = C11×SL2(𝔽3)  order 264 = 23·3·11

### Direct product of C11 and SL2(𝔽3)

Aliases: C11×SL2(𝔽3), Q8⋊C33, C22.A4, (Q8×C11)⋊C3, C2.(C11×A4), SmallGroup(264,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C11×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — Q8×C11 — C11×SL2(𝔽3)
 Lower central Q8 — C11×SL2(𝔽3)
 Upper central C1 — C22

Generators and relations for C11×SL2(𝔽3)
G = < a,b,c,d | a11=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Smallest permutation representation of C11×SL2(𝔽3)
On 88 points
Generators in S88
(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)(45 46 47 48 49 50 51 52 53 54 55)(56 57 58 59 60 61 62 63 64 65 66)(67 68 69 70 71 72 73 74 75 76 77)(78 79 80 81 82 83 84 85 86 87 88)
(1 68 62 80)(2 69 63 81)(3 70 64 82)(4 71 65 83)(5 72 66 84)(6 73 56 85)(7 74 57 86)(8 75 58 87)(9 76 59 88)(10 77 60 78)(11 67 61 79)(12 39 51 25)(13 40 52 26)(14 41 53 27)(15 42 54 28)(16 43 55 29)(17 44 45 30)(18 34 46 31)(19 35 47 32)(20 36 48 33)(21 37 49 23)(22 38 50 24)
(1 32 62 35)(2 33 63 36)(3 23 64 37)(4 24 65 38)(5 25 66 39)(6 26 56 40)(7 27 57 41)(8 28 58 42)(9 29 59 43)(10 30 60 44)(11 31 61 34)(12 72 51 84)(13 73 52 85)(14 74 53 86)(15 75 54 87)(16 76 55 88)(17 77 45 78)(18 67 46 79)(19 68 47 80)(20 69 48 81)(21 70 49 82)(22 71 50 83)
(12 39 84)(13 40 85)(14 41 86)(15 42 87)(16 43 88)(17 44 78)(18 34 79)(19 35 80)(20 36 81)(21 37 82)(22 38 83)(23 70 49)(24 71 50)(25 72 51)(26 73 52)(27 74 53)(28 75 54)(29 76 55)(30 77 45)(31 67 46)(32 68 47)(33 69 48)

G:=sub<Sym(88)| (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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88), (1,68,62,80)(2,69,63,81)(3,70,64,82)(4,71,65,83)(5,72,66,84)(6,73,56,85)(7,74,57,86)(8,75,58,87)(9,76,59,88)(10,77,60,78)(11,67,61,79)(12,39,51,25)(13,40,52,26)(14,41,53,27)(15,42,54,28)(16,43,55,29)(17,44,45,30)(18,34,46,31)(19,35,47,32)(20,36,48,33)(21,37,49,23)(22,38,50,24), (1,32,62,35)(2,33,63,36)(3,23,64,37)(4,24,65,38)(5,25,66,39)(6,26,56,40)(7,27,57,41)(8,28,58,42)(9,29,59,43)(10,30,60,44)(11,31,61,34)(12,72,51,84)(13,73,52,85)(14,74,53,86)(15,75,54,87)(16,76,55,88)(17,77,45,78)(18,67,46,79)(19,68,47,80)(20,69,48,81)(21,70,49,82)(22,71,50,83), (12,39,84)(13,40,85)(14,41,86)(15,42,87)(16,43,88)(17,44,78)(18,34,79)(19,35,80)(20,36,81)(21,37,82)(22,38,83)(23,70,49)(24,71,50)(25,72,51)(26,73,52)(27,74,53)(28,75,54)(29,76,55)(30,77,45)(31,67,46)(32,68,47)(33,69,48)>;

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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66)(67,68,69,70,71,72,73,74,75,76,77)(78,79,80,81,82,83,84,85,86,87,88), (1,68,62,80)(2,69,63,81)(3,70,64,82)(4,71,65,83)(5,72,66,84)(6,73,56,85)(7,74,57,86)(8,75,58,87)(9,76,59,88)(10,77,60,78)(11,67,61,79)(12,39,51,25)(13,40,52,26)(14,41,53,27)(15,42,54,28)(16,43,55,29)(17,44,45,30)(18,34,46,31)(19,35,47,32)(20,36,48,33)(21,37,49,23)(22,38,50,24), (1,32,62,35)(2,33,63,36)(3,23,64,37)(4,24,65,38)(5,25,66,39)(6,26,56,40)(7,27,57,41)(8,28,58,42)(9,29,59,43)(10,30,60,44)(11,31,61,34)(12,72,51,84)(13,73,52,85)(14,74,53,86)(15,75,54,87)(16,76,55,88)(17,77,45,78)(18,67,46,79)(19,68,47,80)(20,69,48,81)(21,70,49,82)(22,71,50,83), (12,39,84)(13,40,85)(14,41,86)(15,42,87)(16,43,88)(17,44,78)(18,34,79)(19,35,80)(20,36,81)(21,37,82)(22,38,83)(23,70,49)(24,71,50)(25,72,51)(26,73,52)(27,74,53)(28,75,54)(29,76,55)(30,77,45)(31,67,46)(32,68,47)(33,69,48) );

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),(45,46,47,48,49,50,51,52,53,54,55),(56,57,58,59,60,61,62,63,64,65,66),(67,68,69,70,71,72,73,74,75,76,77),(78,79,80,81,82,83,84,85,86,87,88)], [(1,68,62,80),(2,69,63,81),(3,70,64,82),(4,71,65,83),(5,72,66,84),(6,73,56,85),(7,74,57,86),(8,75,58,87),(9,76,59,88),(10,77,60,78),(11,67,61,79),(12,39,51,25),(13,40,52,26),(14,41,53,27),(15,42,54,28),(16,43,55,29),(17,44,45,30),(18,34,46,31),(19,35,47,32),(20,36,48,33),(21,37,49,23),(22,38,50,24)], [(1,32,62,35),(2,33,63,36),(3,23,64,37),(4,24,65,38),(5,25,66,39),(6,26,56,40),(7,27,57,41),(8,28,58,42),(9,29,59,43),(10,30,60,44),(11,31,61,34),(12,72,51,84),(13,73,52,85),(14,74,53,86),(15,75,54,87),(16,76,55,88),(17,77,45,78),(18,67,46,79),(19,68,47,80),(20,69,48,81),(21,70,49,82),(22,71,50,83)], [(12,39,84),(13,40,85),(14,41,86),(15,42,87),(16,43,88),(17,44,78),(18,34,79),(19,35,80),(20,36,81),(21,37,82),(22,38,83),(23,70,49),(24,71,50),(25,72,51),(26,73,52),(27,74,53),(28,75,54),(29,76,55),(30,77,45),(31,67,46),(32,68,47),(33,69,48)])

77 conjugacy classes

 class 1 2 3A 3B 4 6A 6B 11A ··· 11J 22A ··· 22J 33A ··· 33T 44A ··· 44J 66A ··· 66T order 1 2 3 3 4 6 6 11 ··· 11 22 ··· 22 33 ··· 33 44 ··· 44 66 ··· 66 size 1 1 4 4 6 4 4 1 ··· 1 1 ··· 1 4 ··· 4 6 ··· 6 4 ··· 4

77 irreducible representations

 dim 1 1 1 1 2 2 2 3 3 type + - + image C1 C3 C11 C33 SL2(𝔽3) SL2(𝔽3) C11×SL2(𝔽3) A4 C11×A4 kernel C11×SL2(𝔽3) Q8×C11 SL2(𝔽3) Q8 C11 C11 C1 C22 C2 # reps 1 2 10 20 1 2 30 1 10

Matrix representation of C11×SL2(𝔽3) in GL2(𝔽23) generated by

 18 0 0 18
,
 20 10 22 3
,
 0 17 4 0
,
 21 7 16 1
G:=sub<GL(2,GF(23))| [18,0,0,18],[20,22,10,3],[0,4,17,0],[21,16,7,1] >;

C11×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_{11}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C11xSL(2,3)");
// GroupNames label

G:=SmallGroup(264,12);
// by ID

G=gap.SmallGroup(264,12);
# by ID

G:=PCGroup([5,-3,-11,-2,2,-2,992,72,1983,133,58]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations

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