direct product, metabelian, soluble, monomial, A-group
Aliases: C13×C32⋊C4, C32⋊C52, C3⋊S3.C26, (C3×C39)⋊5C4, (C13×C3⋊S3).1C2, SmallGroup(468,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C13×C32⋊C4 |
Generators and relations for C13×C32⋊C4
G = < a,b,c,d | a13=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Smallest permutation representation of C13×C32⋊C4
►On 78 points
Generators in S78
(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)
(1 72 50)(2 73 51)(3 74 52)(4 75 40)(5 76 41)(6 77 42)(7 78 43)(8 66 44)(9 67 45)(10 68 46)(11 69 47)(12 70 48)(13 71 49)(14 57 34)(15 58 35)(16 59 36)(17 60 37)(18 61 38)(19 62 39)(20 63 27)(21 64 28)(22 65 29)(23 53 30)(24 54 31)(25 55 32)(26 56 33)
(14 34 57)(15 35 58)(16 36 59)(17 37 60)(18 38 61)(19 39 62)(20 27 63)(21 28 64)(22 29 65)(23 30 53)(24 31 54)(25 32 55)(26 33 56)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 73 57 51)(15 74 58 52)(16 75 59 40)(17 76 60 41)(18 77 61 42)(19 78 62 43)(20 66 63 44)(21 67 64 45)(22 68 65 46)(23 69 53 47)(24 70 54 48)(25 71 55 49)(26 72 56 50)
G:=sub<Sym(78)| (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), (1,72,50)(2,73,51)(3,74,52)(4,75,40)(5,76,41)(6,77,42)(7,78,43)(8,66,44)(9,67,45)(10,68,46)(11,69,47)(12,70,48)(13,71,49)(14,57,34)(15,58,35)(16,59,36)(17,60,37)(18,61,38)(19,62,39)(20,63,27)(21,64,28)(22,65,29)(23,53,30)(24,54,31)(25,55,32)(26,56,33), (14,34,57)(15,35,58)(16,36,59)(17,37,60)(18,38,61)(19,39,62)(20,27,63)(21,28,64)(22,29,65)(23,30,53)(24,31,54)(25,32,55)(26,33,56), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,73,57,51)(15,74,58,52)(16,75,59,40)(17,76,60,41)(18,77,61,42)(19,78,62,43)(20,66,63,44)(21,67,64,45)(22,68,65,46)(23,69,53,47)(24,70,54,48)(25,71,55,49)(26,72,56,50)>;
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), (1,72,50)(2,73,51)(3,74,52)(4,75,40)(5,76,41)(6,77,42)(7,78,43)(8,66,44)(9,67,45)(10,68,46)(11,69,47)(12,70,48)(13,71,49)(14,57,34)(15,58,35)(16,59,36)(17,60,37)(18,61,38)(19,62,39)(20,63,27)(21,64,28)(22,65,29)(23,53,30)(24,54,31)(25,55,32)(26,56,33), (14,34,57)(15,35,58)(16,36,59)(17,37,60)(18,38,61)(19,39,62)(20,27,63)(21,28,64)(22,29,65)(23,30,53)(24,31,54)(25,32,55)(26,33,56), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,73,57,51)(15,74,58,52)(16,75,59,40)(17,76,60,41)(18,77,61,42)(19,78,62,43)(20,66,63,44)(21,67,64,45)(22,68,65,46)(23,69,53,47)(24,70,54,48)(25,71,55,49)(26,72,56,50) );
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)], [(1,72,50),(2,73,51),(3,74,52),(4,75,40),(5,76,41),(6,77,42),(7,78,43),(8,66,44),(9,67,45),(10,68,46),(11,69,47),(12,70,48),(13,71,49),(14,57,34),(15,58,35),(16,59,36),(17,60,37),(18,61,38),(19,62,39),(20,63,27),(21,64,28),(22,65,29),(23,53,30),(24,54,31),(25,55,32),(26,56,33)], [(14,34,57),(15,35,58),(16,36,59),(17,37,60),(18,38,61),(19,39,62),(20,27,63),(21,28,64),(22,29,65),(23,30,53),(24,31,54),(25,32,55),(26,33,56)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,73,57,51),(15,74,58,52),(16,75,59,40),(17,76,60,41),(18,77,61,42),(19,78,62,43),(20,66,63,44),(21,67,64,45),(22,68,65,46),(23,69,53,47),(24,70,54,48),(25,71,55,49),(26,72,56,50)]])
78 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39X | 52A | ··· | 52X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 |
| size | 1 | 9 | 4 | 4 | 9 | 9 | 1 | ··· | 1 | 9 | ··· | 9 | 4 | ··· | 4 | 9 | ··· | 9 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C13 | C26 | C52 | C32⋊C4 | C13×C32⋊C4 |
| kernel | C13×C32⋊C4 | C13×C3⋊S3 | C3×C39 | C32⋊C4 | C3⋊S3 | C32 | C13 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 | 2 | 24 |
Matrix representation of C13×C32⋊C4 ►in GL4(𝔽157) generated by
| 39 | 0 | 0 | 0 |
| 0 | 39 | 0 | 0 |
| 0 | 0 | 39 | 0 |
| 0 | 0 | 0 | 39 |
| 156 | 156 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 156 | 156 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 156 | 156 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 156 | 156 | 0 | 0 |
G:=sub<GL(4,GF(157))| [39,0,0,0,0,39,0,0,0,0,39,0,0,0,0,39],[156,1,0,0,156,0,0,0,0,0,156,1,0,0,156,0],[1,0,0,0,0,1,0,0,0,0,0,156,0,0,1,156],[0,0,1,156,0,0,0,156,1,0,0,0,0,1,0,0] >;
C13×C32⋊C4 in GAP, Magma, Sage, TeX
C_{13}\times C_3^2\rtimes C_4 % in TeX
G:=Group("C13xC3^2:C4"); // GroupNames label
G:=SmallGroup(468,39);
// by ID
G=gap.SmallGroup(468,39);
# by ID
G:=PCGroup([5,-2,-13,-2,-3,3,130,7283,93,10404,314]);
// Polycyclic
G:=Group<a,b,c,d|a^13=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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