metabelian, soluble, monomial, A-group
Aliases: C32⋊Dic11, (C3×C33)⋊2C4, C11⋊(C32⋊C4), C3⋊S3.D11, (C11×C3⋊S3).2C2, SmallGroup(396,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C33 — C32⋊Dic11 |
Generators and relations for C32⋊Dic11
G = < a,b,c,d | a3=b3=c22=1, d2=c11, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >
Smallest permutation representation of C32⋊Dic11
►On 66 points
Generators in S66
(1 25 36)(2 37 26)(3 27 38)(4 39 28)(5 29 40)(6 41 30)(7 31 42)(8 43 32)(9 33 44)(10 23 34)(11 35 24)(12 62 51)(13 52 63)(14 64 53)(15 54 65)(16 66 55)(17 56 45)(18 46 57)(19 58 47)(20 48 59)(21 60 49)(22 50 61)
(12 51 62)(13 63 52)(14 53 64)(15 65 54)(16 55 66)(17 45 56)(18 57 46)(19 47 58)(20 59 48)(21 49 60)(22 61 50)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 22)(8 21)(9 20)(10 19)(11 18)(23 58 34 47)(24 57 35 46)(25 56 36 45)(26 55 37 66)(27 54 38 65)(28 53 39 64)(29 52 40 63)(30 51 41 62)(31 50 42 61)(32 49 43 60)(33 48 44 59)
G:=sub<Sym(66)| (1,25,36)(2,37,26)(3,27,38)(4,39,28)(5,29,40)(6,41,30)(7,31,42)(8,43,32)(9,33,44)(10,23,34)(11,35,24)(12,62,51)(13,52,63)(14,64,53)(15,54,65)(16,66,55)(17,56,45)(18,46,57)(19,58,47)(20,48,59)(21,60,49)(22,50,61), (12,51,62)(13,63,52)(14,53,64)(15,65,54)(16,55,66)(17,45,56)(18,57,46)(19,47,58)(20,59,48)(21,49,60)(22,61,50), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,22)(8,21)(9,20)(10,19)(11,18)(23,58,34,47)(24,57,35,46)(25,56,36,45)(26,55,37,66)(27,54,38,65)(28,53,39,64)(29,52,40,63)(30,51,41,62)(31,50,42,61)(32,49,43,60)(33,48,44,59)>;
G:=Group( (1,25,36)(2,37,26)(3,27,38)(4,39,28)(5,29,40)(6,41,30)(7,31,42)(8,43,32)(9,33,44)(10,23,34)(11,35,24)(12,62,51)(13,52,63)(14,64,53)(15,54,65)(16,66,55)(17,56,45)(18,46,57)(19,58,47)(20,48,59)(21,60,49)(22,50,61), (12,51,62)(13,63,52)(14,53,64)(15,65,54)(16,55,66)(17,45,56)(18,57,46)(19,47,58)(20,59,48)(21,49,60)(22,61,50), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,22)(8,21)(9,20)(10,19)(11,18)(23,58,34,47)(24,57,35,46)(25,56,36,45)(26,55,37,66)(27,54,38,65)(28,53,39,64)(29,52,40,63)(30,51,41,62)(31,50,42,61)(32,49,43,60)(33,48,44,59) );
G=PermutationGroup([[(1,25,36),(2,37,26),(3,27,38),(4,39,28),(5,29,40),(6,41,30),(7,31,42),(8,43,32),(9,33,44),(10,23,34),(11,35,24),(12,62,51),(13,52,63),(14,64,53),(15,54,65),(16,66,55),(17,56,45),(18,46,57),(19,58,47),(20,48,59),(21,60,49),(22,50,61)], [(12,51,62),(13,63,52),(14,53,64),(15,65,54),(16,55,66),(17,45,56),(18,57,46),(19,47,58),(20,59,48),(21,49,60),(22,61,50)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,22),(8,21),(9,20),(10,19),(11,18),(23,58,34,47),(24,57,35,46),(25,56,36,45),(26,55,37,66),(27,54,38,65),(28,53,39,64),(29,52,40,63),(30,51,41,62),(31,50,42,61),(32,49,43,60),(33,48,44,59)]])
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 11A | ··· | 11E | 22A | ··· | 22E | 33A | ··· | 33T |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 |
| size | 1 | 9 | 4 | 4 | 99 | 99 | 2 | ··· | 2 | 18 | ··· | 18 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | D11 | Dic11 | C32⋊C4 | C32⋊Dic11 |
| kernel | C32⋊Dic11 | C11×C3⋊S3 | C3×C33 | C3⋊S3 | C32 | C11 | C1 |
| # reps | 1 | 1 | 2 | 5 | 5 | 2 | 20 |
Matrix representation of C32⋊Dic11 ►in GL4(𝔽397) generated by
| 0 | 1 | 0 | 0 |
| 396 | 396 | 0 | 0 |
| 0 | 0 | 396 | 396 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 396 | 396 |
| 393 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 99 | 0 |
| 0 | 0 | 298 | 298 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 396 | 396 | 0 | 0 |
G:=sub<GL(4,GF(397))| [0,396,0,0,1,396,0,0,0,0,396,1,0,0,396,0],[1,0,0,0,0,1,0,0,0,0,0,396,0,0,1,396],[393,4,0,0,0,4,0,0,0,0,99,298,0,0,0,298],[0,0,1,396,0,0,0,396,1,0,0,0,0,1,0,0] >;
C32⋊Dic11 in GAP, Magma, Sage, TeX
C_3^2\rtimes {\rm Dic}_{11} % in TeX
G:=Group("C3^2:Dic11"); // GroupNames label
G:=SmallGroup(396,18);
// by ID
G=gap.SmallGroup(396,18);
# by ID
G:=PCGroup([5,-2,-2,-3,3,-11,10,302,67,323,248,9004]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^22=1,d^2=c^11,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
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