direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×S3×D11, C33⋊C23, C22⋊1D6, C6⋊1D22, C66⋊C22, D66⋊5C2, D33⋊C22, (S3×C22)⋊3C2, (C6×D11)⋊3C2, (S3×C11)⋊C22, C11⋊1(C22×S3), (C3×D11)⋊C22, C3⋊1(C22×D11), SmallGroup(264,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C2×S3×D11 |
Generators and relations for C2×S3×D11
G = < a,b,c,d,e | a2=b3=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 488 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C22, S3, S3, C6, C6, C23, C11, D6, D6, C2×C6, D11, D11, C22, C22, C22×S3, C33, D22, D22, C2×C22, S3×C11, C3×D11, D33, C66, C22×D11, S3×D11, C6×D11, S3×C22, D66, C2×S3×D11
Quotients: C1, C2, C22, S3, C23, D6, D11, C22×S3, D22, C22×D11, S3×D11, C2×S3×D11
►On 66 points
Generators in S66
(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)
(1 21 32)(2 22 33)(3 12 23)(4 13 24)(5 14 25)(6 15 26)(7 16 27)(8 17 28)(9 18 29)(10 19 30)(11 20 31)(34 45 56)(35 46 57)(36 47 58)(37 48 59)(38 49 60)(39 50 61)(40 51 62)(41 52 63)(42 53 64)(43 54 65)(44 55 66)
(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)
(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)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)
G:=sub<Sym(66)| (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66), (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31)(34,45,56)(35,46,57)(36,47,58)(37,48,59)(38,49,60)(39,50,61)(40,51,62)(41,52,63)(42,53,64)(43,54,65)(44,55,66), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)>;
G:=Group( (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66), (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31)(34,45,56)(35,46,57)(36,47,58)(37,48,59)(38,49,60)(39,50,61)(40,51,62)(41,52,63)(42,53,64)(43,54,65)(44,55,66), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65) );
G=PermutationGroup([[(1,43),(2,44),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66)], [(1,21,32),(2,22,33),(3,12,23),(4,13,24),(5,14,25),(6,15,26),(7,16,27),(8,17,28),(9,18,29),(10,19,30),(11,20,31),(34,45,56),(35,46,57),(36,47,58),(37,48,59),(38,49,60),(39,50,61),(40,51,62),(41,52,63),(42,53,64),(43,54,65),(44,55,66)], [(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22O | 33A | ··· | 33E | 66A | ··· | 66E |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 3 | 3 | 11 | 11 | 33 | 33 | 2 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D11 | D22 | D22 | S3×D11 | C2×S3×D11 |
| kernel | C2×S3×D11 | S3×D11 | C6×D11 | S3×C22 | D66 | D22 | D11 | C22 | D6 | S3 | C6 | C2 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 5 | 10 | 5 | 5 | 5 |
Matrix representation of C2×S3×D11 ►in GL4(𝔽67) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 0 |
| 0 | 0 | 0 | 66 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 65 | 17 |
| 0 | 0 | 55 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 66 |
| 0 | 1 | 0 | 0 |
| 66 | 19 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(67))| [1,0,0,0,0,1,0,0,0,0,66,0,0,0,0,66],[1,0,0,0,0,1,0,0,0,0,65,55,0,0,17,1],[1,0,0,0,0,1,0,0,0,0,1,12,0,0,0,66],[0,66,0,0,1,19,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
C2×S3×D11 in GAP, Magma, Sage, TeX
C_2\times S_3\times D_{11} % in TeX
G:=Group("C2xS3xD11"); // GroupNames label
G:=SmallGroup(264,34);
// by ID
G=gap.SmallGroup(264,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,168,6004]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^11=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations