direct product, non-abelian, soluble
Aliases: C11×SL2(𝔽3), Q8⋊C33, C22.A4, (Q8×C11)⋊C3, C2.(C11×A4), SmallGroup(264,12)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C11×SL2(𝔽3) |
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 30 80)(2 69 31 81)(3 70 32 82)(4 71 33 83)(5 72 23 84)(6 73 24 85)(7 74 25 86)(8 75 26 87)(9 76 27 88)(10 77 28 78)(11 67 29 79)(12 53 44 61)(13 54 34 62)(14 55 35 63)(15 45 36 64)(16 46 37 65)(17 47 38 66)(18 48 39 56)(19 49 40 57)(20 50 41 58)(21 51 42 59)(22 52 43 60)
(1 14 30 35)(2 15 31 36)(3 16 32 37)(4 17 33 38)(5 18 23 39)(6 19 24 40)(7 20 25 41)(8 21 26 42)(9 22 27 43)(10 12 28 44)(11 13 29 34)(45 69 64 81)(46 70 65 82)(47 71 66 83)(48 72 56 84)(49 73 57 85)(50 74 58 86)(51 75 59 87)(52 76 60 88)(53 77 61 78)(54 67 62 79)(55 68 63 80)
(12 77 61)(13 67 62)(14 68 63)(15 69 64)(16 70 65)(17 71 66)(18 72 56)(19 73 57)(20 74 58)(21 75 59)(22 76 60)(34 79 54)(35 80 55)(36 81 45)(37 82 46)(38 83 47)(39 84 48)(40 85 49)(41 86 50)(42 87 51)(43 88 52)(44 78 53)
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), (1,68,30,80)(2,69,31,81)(3,70,32,82)(4,71,33,83)(5,72,23,84)(6,73,24,85)(7,74,25,86)(8,75,26,87)(9,76,27,88)(10,77,28,78)(11,67,29,79)(12,53,44,61)(13,54,34,62)(14,55,35,63)(15,45,36,64)(16,46,37,65)(17,47,38,66)(18,48,39,56)(19,49,40,57)(20,50,41,58)(21,51,42,59)(22,52,43,60), (1,14,30,35)(2,15,31,36)(3,16,32,37)(4,17,33,38)(5,18,23,39)(6,19,24,40)(7,20,25,41)(8,21,26,42)(9,22,27,43)(10,12,28,44)(11,13,29,34)(45,69,64,81)(46,70,65,82)(47,71,66,83)(48,72,56,84)(49,73,57,85)(50,74,58,86)(51,75,59,87)(52,76,60,88)(53,77,61,78)(54,67,62,79)(55,68,63,80), (12,77,61)(13,67,62)(14,68,63)(15,69,64)(16,70,65)(17,71,66)(18,72,56)(19,73,57)(20,74,58)(21,75,59)(22,76,60)(34,79,54)(35,80,55)(36,81,45)(37,82,46)(38,83,47)(39,84,48)(40,85,49)(41,86,50)(42,87,51)(43,88,52)(44,78,53)>;
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), (1,68,30,80)(2,69,31,81)(3,70,32,82)(4,71,33,83)(5,72,23,84)(6,73,24,85)(7,74,25,86)(8,75,26,87)(9,76,27,88)(10,77,28,78)(11,67,29,79)(12,53,44,61)(13,54,34,62)(14,55,35,63)(15,45,36,64)(16,46,37,65)(17,47,38,66)(18,48,39,56)(19,49,40,57)(20,50,41,58)(21,51,42,59)(22,52,43,60), (1,14,30,35)(2,15,31,36)(3,16,32,37)(4,17,33,38)(5,18,23,39)(6,19,24,40)(7,20,25,41)(8,21,26,42)(9,22,27,43)(10,12,28,44)(11,13,29,34)(45,69,64,81)(46,70,65,82)(47,71,66,83)(48,72,56,84)(49,73,57,85)(50,74,58,86)(51,75,59,87)(52,76,60,88)(53,77,61,78)(54,67,62,79)(55,68,63,80), (12,77,61)(13,67,62)(14,68,63)(15,69,64)(16,70,65)(17,71,66)(18,72,56)(19,73,57)(20,74,58)(21,75,59)(22,76,60)(34,79,54)(35,80,55)(36,81,45)(37,82,46)(38,83,47)(39,84,48)(40,85,49)(41,86,50)(42,87,51)(43,88,52)(44,78,53) );
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)], [(1,68,30,80),(2,69,31,81),(3,70,32,82),(4,71,33,83),(5,72,23,84),(6,73,24,85),(7,74,25,86),(8,75,26,87),(9,76,27,88),(10,77,28,78),(11,67,29,79),(12,53,44,61),(13,54,34,62),(14,55,35,63),(15,45,36,64),(16,46,37,65),(17,47,38,66),(18,48,39,56),(19,49,40,57),(20,50,41,58),(21,51,42,59),(22,52,43,60)], [(1,14,30,35),(2,15,31,36),(3,16,32,37),(4,17,33,38),(5,18,23,39),(6,19,24,40),(7,20,25,41),(8,21,26,42),(9,22,27,43),(10,12,28,44),(11,13,29,34),(45,69,64,81),(46,70,65,82),(47,71,66,83),(48,72,56,84),(49,73,57,85),(50,74,58,86),(51,75,59,87),(52,76,60,88),(53,77,61,78),(54,67,62,79),(55,68,63,80)], [(12,77,61),(13,67,62),(14,68,63),(15,69,64),(16,70,65),(17,71,66),(18,72,56),(19,73,57),(20,74,58),(21,75,59),(22,76,60),(34,79,54),(35,80,55),(36,81,45),(37,82,46),(38,83,47),(39,84,48),(40,85,49),(41,86,50),(42,87,51),(43,88,52),(44,78,53)]])
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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