direct product, metabelian, soluble, monomial, A-group
Aliases: C11×C32⋊C4, C32⋊C44, C3⋊S3.C22, (C3×C33)⋊1C4, (C11×C3⋊S3).1C2, SmallGroup(396,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C11×C32⋊C4 |
Generators and relations for C11×C32⋊C4
G = < a,b,c,d | a11=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Smallest permutation representation of C11×C32⋊C4
►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 53 38)(2 54 39)(3 55 40)(4 45 41)(5 46 42)(6 47 43)(7 48 44)(8 49 34)(9 50 35)(10 51 36)(11 52 37)(12 63 24)(13 64 25)(14 65 26)(15 66 27)(16 56 28)(17 57 29)(18 58 30)(19 59 31)(20 60 32)(21 61 33)(22 62 23)
(12 24 63)(13 25 64)(14 26 65)(15 27 66)(16 28 56)(17 29 57)(18 30 58)(19 31 59)(20 32 60)(21 33 61)(22 23 62)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 12)(7 13)(8 14)(9 15)(10 16)(11 17)(23 42 62 46)(24 43 63 47)(25 44 64 48)(26 34 65 49)(27 35 66 50)(28 36 56 51)(29 37 57 52)(30 38 58 53)(31 39 59 54)(32 40 60 55)(33 41 61 45)
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), (1,53,38)(2,54,39)(3,55,40)(4,45,41)(5,46,42)(6,47,43)(7,48,44)(8,49,34)(9,50,35)(10,51,36)(11,52,37)(12,63,24)(13,64,25)(14,65,26)(15,66,27)(16,56,28)(17,57,29)(18,58,30)(19,59,31)(20,60,32)(21,61,33)(22,62,23), (12,24,63)(13,25,64)(14,26,65)(15,27,66)(16,28,56)(17,29,57)(18,30,58)(19,31,59)(20,32,60)(21,33,61)(22,23,62), (1,18)(2,19)(3,20)(4,21)(5,22)(6,12)(7,13)(8,14)(9,15)(10,16)(11,17)(23,42,62,46)(24,43,63,47)(25,44,64,48)(26,34,65,49)(27,35,66,50)(28,36,56,51)(29,37,57,52)(30,38,58,53)(31,39,59,54)(32,40,60,55)(33,41,61,45)>;
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), (1,53,38)(2,54,39)(3,55,40)(4,45,41)(5,46,42)(6,47,43)(7,48,44)(8,49,34)(9,50,35)(10,51,36)(11,52,37)(12,63,24)(13,64,25)(14,65,26)(15,66,27)(16,56,28)(17,57,29)(18,58,30)(19,59,31)(20,60,32)(21,61,33)(22,62,23), (12,24,63)(13,25,64)(14,26,65)(15,27,66)(16,28,56)(17,29,57)(18,30,58)(19,31,59)(20,32,60)(21,33,61)(22,23,62), (1,18)(2,19)(3,20)(4,21)(5,22)(6,12)(7,13)(8,14)(9,15)(10,16)(11,17)(23,42,62,46)(24,43,63,47)(25,44,64,48)(26,34,65,49)(27,35,66,50)(28,36,56,51)(29,37,57,52)(30,38,58,53)(31,39,59,54)(32,40,60,55)(33,41,61,45) );
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)], [(1,53,38),(2,54,39),(3,55,40),(4,45,41),(5,46,42),(6,47,43),(7,48,44),(8,49,34),(9,50,35),(10,51,36),(11,52,37),(12,63,24),(13,64,25),(14,65,26),(15,66,27),(16,56,28),(17,57,29),(18,58,30),(19,59,31),(20,60,32),(21,61,33),(22,62,23)], [(12,24,63),(13,25,64),(14,26,65),(15,27,66),(16,28,56),(17,29,57),(18,30,58),(19,31,59),(20,32,60),(21,33,61),(22,23,62)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,12),(7,13),(8,14),(9,15),(10,16),(11,17),(23,42,62,46),(24,43,63,47),(25,44,64,48),(26,34,65,49),(27,35,66,50),(28,36,56,51),(29,37,57,52),(30,38,58,53),(31,39,59,54),(32,40,60,55),(33,41,61,45)]])
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 11A | ··· | 11J | 22A | ··· | 22J | 33A | ··· | 33T | 44A | ··· | 44T |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 |
| size | 1 | 9 | 4 | 4 | 9 | 9 | 1 | ··· | 1 | 9 | ··· | 9 | 4 | ··· | 4 | 9 | ··· | 9 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C11 | C22 | C44 | C32⋊C4 | C11×C32⋊C4 |
| kernel | C11×C32⋊C4 | C11×C3⋊S3 | C3×C33 | C32⋊C4 | C3⋊S3 | C32 | C11 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 2 | 20 |
Matrix representation of C11×C32⋊C4 ►in GL4(𝔽397) generated by
| 333 | 0 | 0 | 0 |
| 0 | 333 | 0 | 0 |
| 0 | 0 | 333 | 0 |
| 0 | 0 | 0 | 333 |
| 396 | 396 | 267 | 0 |
| 1 | 0 | 0 | 130 |
| 0 | 0 | 0 | 396 |
| 0 | 0 | 1 | 396 |
| 1 | 0 | 130 | 0 |
| 0 | 1 | 130 | 0 |
| 0 | 0 | 396 | 1 |
| 0 | 0 | 396 | 0 |
| 130 | 0 | 75 | 76 |
| 130 | 0 | 75 | 75 |
| 395 | 396 | 267 | 267 |
| 396 | 1 | 0 | 0 |
G:=sub<GL(4,GF(397))| [333,0,0,0,0,333,0,0,0,0,333,0,0,0,0,333],[396,1,0,0,396,0,0,0,267,0,0,1,0,130,396,396],[1,0,0,0,0,1,0,0,130,130,396,396,0,0,1,0],[130,130,395,396,0,0,396,1,75,75,267,0,76,75,267,0] >;
C11×C32⋊C4 in GAP, Magma, Sage, TeX
C_{11}\times C_3^2\rtimes C_4 % in TeX
G:=Group("C11xC3^2:C4"); // GroupNames label
G:=SmallGroup(396,17);
// by ID
G=gap.SmallGroup(396,17);
# by ID
G:=PCGroup([5,-2,-11,-2,-3,3,110,6163,93,8804,314]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
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