direct product, metabelian, supersoluble, monomial, A-group
Aliases: S32×C11, C33⋊5D6, C3⋊S3⋊C22, (C3×S3)⋊C22, C32⋊(C2×C22), C3⋊1(S3×C22), (S3×C33)⋊3C2, (C3×C33)⋊5C22, (C11×C3⋊S3)⋊3C2, SmallGroup(396,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — S32×C11 |
Generators and relations for S32×C11
G = < a,b,c,d,e | a11=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Smallest permutation representation of S32×C11
►On 66 points
Generators in S66
(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 55 38)(2 45 39)(3 46 40)(4 47 41)(5 48 42)(6 49 43)(7 50 44)(8 51 34)(9 52 35)(10 53 36)(11 54 37)(12 61 33)(13 62 23)(14 63 24)(15 64 25)(16 65 26)(17 66 27)(18 56 28)(19 57 29)(20 58 30)(21 59 31)(22 60 32)
(1 20)(2 21)(3 22)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 45)(32 46)(33 47)(34 65)(35 66)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)
(1 38 55)(2 39 45)(3 40 46)(4 41 47)(5 42 48)(6 43 49)(7 44 50)(8 34 51)(9 35 52)(10 36 53)(11 37 54)(12 61 33)(13 62 23)(14 63 24)(15 64 25)(16 65 26)(17 66 27)(18 56 28)(19 57 29)(20 58 30)(21 59 31)(22 60 32)
(1 20)(2 21)(3 22)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(23 42)(24 43)(25 44)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(33 41)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 56)(54 57)(55 58)
G:=sub<Sym(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), (1,55,38)(2,45,39)(3,46,40)(4,47,41)(5,48,42)(6,49,43)(7,50,44)(8,51,34)(9,52,35)(10,53,36)(11,54,37)(12,61,33)(13,62,23)(14,63,24)(15,64,25)(16,65,26)(17,66,27)(18,56,28)(19,57,29)(20,58,30)(21,59,31)(22,60,32), (1,20)(2,21)(3,22)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,45)(32,46)(33,47)(34,65)(35,66)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64), (1,38,55)(2,39,45)(3,40,46)(4,41,47)(5,42,48)(6,43,49)(7,44,50)(8,34,51)(9,35,52)(10,36,53)(11,37,54)(12,61,33)(13,62,23)(14,63,24)(15,64,25)(16,65,26)(17,66,27)(18,56,28)(19,57,29)(20,58,30)(21,59,31)(22,60,32), (1,20)(2,21)(3,22)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(23,42)(24,43)(25,44)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(33,41)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,56)(54,57)(55,58)>;
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), (1,55,38)(2,45,39)(3,46,40)(4,47,41)(5,48,42)(6,49,43)(7,50,44)(8,51,34)(9,52,35)(10,53,36)(11,54,37)(12,61,33)(13,62,23)(14,63,24)(15,64,25)(16,65,26)(17,66,27)(18,56,28)(19,57,29)(20,58,30)(21,59,31)(22,60,32), (1,20)(2,21)(3,22)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,45)(32,46)(33,47)(34,65)(35,66)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64), (1,38,55)(2,39,45)(3,40,46)(4,41,47)(5,42,48)(6,43,49)(7,44,50)(8,34,51)(9,35,52)(10,36,53)(11,37,54)(12,61,33)(13,62,23)(14,63,24)(15,64,25)(16,65,26)(17,66,27)(18,56,28)(19,57,29)(20,58,30)(21,59,31)(22,60,32), (1,20)(2,21)(3,22)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(23,42)(24,43)(25,44)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(33,41)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,56)(54,57)(55,58) );
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)], [(1,55,38),(2,45,39),(3,46,40),(4,47,41),(5,48,42),(6,49,43),(7,50,44),(8,51,34),(9,52,35),(10,53,36),(11,54,37),(12,61,33),(13,62,23),(14,63,24),(15,64,25),(16,65,26),(17,66,27),(18,56,28),(19,57,29),(20,58,30),(21,59,31),(22,60,32)], [(1,20),(2,21),(3,22),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,45),(32,46),(33,47),(34,65),(35,66),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64)], [(1,38,55),(2,39,45),(3,40,46),(4,41,47),(5,42,48),(6,43,49),(7,44,50),(8,34,51),(9,35,52),(10,36,53),(11,37,54),(12,61,33),(13,62,23),(14,63,24),(15,64,25),(16,65,26),(17,66,27),(18,56,28),(19,57,29),(20,58,30),(21,59,31),(22,60,32)], [(1,20),(2,21),(3,22),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(23,42),(24,43),(25,44),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40),(33,41),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,56),(54,57),(55,58)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 6A | 6B | 11A | ··· | 11J | 22A | ··· | 22T | 22U | ··· | 22AD | 33A | ··· | 33T | 33U | ··· | 33AD | 66A | ··· | 66T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 3 | 3 | 9 | 2 | 2 | 4 | 6 | 6 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | S3 | D6 | S3×C11 | S3×C22 | S32 | S32×C11 |
| kernel | S32×C11 | S3×C33 | C11×C3⋊S3 | S32 | C3×S3 | C3⋊S3 | S3×C11 | C33 | S3 | C3 | C11 | C1 |
| # reps | 1 | 2 | 1 | 10 | 20 | 10 | 2 | 2 | 20 | 20 | 1 | 10 |
Matrix representation of S32×C11 ►in GL4(𝔽67) generated by
| 14 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 1 |
| 0 | 0 | 66 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 66 | 66 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 66 | 66 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(67))| [14,0,0,0,0,14,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,66,66,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,66,0,0,1,66,0,0,0,0,1,0,0,0,0,1],[1,66,0,0,0,66,0,0,0,0,1,0,0,0,0,1] >;
S32×C11 in GAP, Magma, Sage, TeX
S_3^2\times C_{11} % in TeX
G:=Group("S3^2xC11"); // GroupNames label
G:=SmallGroup(396,21);
// by ID
G=gap.SmallGroup(396,21);
# by ID
G:=PCGroup([5,-2,-2,-11,-3,-3,888,6604]);
// Polycyclic
G:=Group<a,b,c,d,e|a^11=b^3=c^2=d^3=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
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