non-abelian, supersoluble, monomial
Aliases: C34⋊13S3, He3⋊5(C3⋊S3), C3⋊(He3⋊5S3), (C3×He3)⋊24S3, (C32×He3)⋊8C2, C33⋊10(C3⋊S3), C3.2(C34⋊C2), C32⋊4(He3⋊C2), C32⋊2(C33⋊C2), SmallGroup(486,248)
Series: Derived ►Chief ►Lower central ►Upper central
| C32×He3 — C34⋊13S3 |
Generators and relations for C34⋊13S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=cd=dc, fcf=c-1, de=ed, df=fd, fef=e-1 >
Subgroups: 4948 in 684 conjugacy classes, 219 normal (6 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C32, C3×S3, C3⋊S3, He3, C33, C33, C33, He3⋊C2, C3×C3⋊S3, C33⋊C2, C3×He3, C34, He3⋊5S3, C3×C33⋊C2, C32×He3, C34⋊13S3
Quotients: C1, C2, S3, C3⋊S3, He3⋊C2, C33⋊C2, He3⋊5S3, C34⋊C2, C34⋊13S3
►On 54 points
Generators in S54
(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)
(1 13 34)(2 14 35)(3 15 36)(4 26 12)(5 27 10)(6 25 11)(7 19 41)(8 20 42)(9 21 40)(16 50 47)(17 51 48)(18 49 46)(22 29 45)(23 30 43)(24 28 44)(31 38 52)(32 39 53)(33 37 54)
(1 25 20)(2 26 21)(3 27 19)(4 9 35)(5 7 36)(6 8 34)(10 41 15)(11 42 13)(12 40 14)(16 37 28)(17 38 29)(18 39 30)(22 48 31)(23 46 32)(24 47 33)(43 49 53)(44 50 54)(45 51 52)
(1 15 35)(2 13 36)(3 14 34)(4 25 10)(5 26 11)(6 27 12)(7 21 42)(8 19 40)(9 20 41)(16 48 49)(17 46 50)(18 47 51)(22 43 28)(23 44 29)(24 45 30)(31 53 37)(32 54 38)(33 52 39)
(1 7 6)(2 8 4)(3 9 5)(10 36 40)(11 34 41)(12 35 42)(13 19 25)(14 20 26)(15 21 27)(16 23 39)(17 24 37)(18 22 38)(28 54 51)(29 52 49)(30 53 50)(31 46 45)(32 47 43)(33 48 44)
(1 43)(2 45)(3 44)(4 31)(5 33)(6 32)(7 47)(8 46)(9 48)(10 37)(11 39)(12 38)(13 30)(14 29)(15 28)(16 41)(17 40)(18 42)(19 50)(20 49)(21 51)(22 35)(23 34)(24 36)(25 53)(26 52)(27 54)
G:=sub<Sym(54)| (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), (1,13,34)(2,14,35)(3,15,36)(4,26,12)(5,27,10)(6,25,11)(7,19,41)(8,20,42)(9,21,40)(16,50,47)(17,51,48)(18,49,46)(22,29,45)(23,30,43)(24,28,44)(31,38,52)(32,39,53)(33,37,54), (1,25,20)(2,26,21)(3,27,19)(4,9,35)(5,7,36)(6,8,34)(10,41,15)(11,42,13)(12,40,14)(16,37,28)(17,38,29)(18,39,30)(22,48,31)(23,46,32)(24,47,33)(43,49,53)(44,50,54)(45,51,52), (1,15,35)(2,13,36)(3,14,34)(4,25,10)(5,26,11)(6,27,12)(7,21,42)(8,19,40)(9,20,41)(16,48,49)(17,46,50)(18,47,51)(22,43,28)(23,44,29)(24,45,30)(31,53,37)(32,54,38)(33,52,39), (1,7,6)(2,8,4)(3,9,5)(10,36,40)(11,34,41)(12,35,42)(13,19,25)(14,20,26)(15,21,27)(16,23,39)(17,24,37)(18,22,38)(28,54,51)(29,52,49)(30,53,50)(31,46,45)(32,47,43)(33,48,44), (1,43)(2,45)(3,44)(4,31)(5,33)(6,32)(7,47)(8,46)(9,48)(10,37)(11,39)(12,38)(13,30)(14,29)(15,28)(16,41)(17,40)(18,42)(19,50)(20,49)(21,51)(22,35)(23,34)(24,36)(25,53)(26,52)(27,54)>;
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), (1,13,34)(2,14,35)(3,15,36)(4,26,12)(5,27,10)(6,25,11)(7,19,41)(8,20,42)(9,21,40)(16,50,47)(17,51,48)(18,49,46)(22,29,45)(23,30,43)(24,28,44)(31,38,52)(32,39,53)(33,37,54), (1,25,20)(2,26,21)(3,27,19)(4,9,35)(5,7,36)(6,8,34)(10,41,15)(11,42,13)(12,40,14)(16,37,28)(17,38,29)(18,39,30)(22,48,31)(23,46,32)(24,47,33)(43,49,53)(44,50,54)(45,51,52), (1,15,35)(2,13,36)(3,14,34)(4,25,10)(5,26,11)(6,27,12)(7,21,42)(8,19,40)(9,20,41)(16,48,49)(17,46,50)(18,47,51)(22,43,28)(23,44,29)(24,45,30)(31,53,37)(32,54,38)(33,52,39), (1,7,6)(2,8,4)(3,9,5)(10,36,40)(11,34,41)(12,35,42)(13,19,25)(14,20,26)(15,21,27)(16,23,39)(17,24,37)(18,22,38)(28,54,51)(29,52,49)(30,53,50)(31,46,45)(32,47,43)(33,48,44), (1,43)(2,45)(3,44)(4,31)(5,33)(6,32)(7,47)(8,46)(9,48)(10,37)(11,39)(12,38)(13,30)(14,29)(15,28)(16,41)(17,40)(18,42)(19,50)(20,49)(21,51)(22,35)(23,34)(24,36)(25,53)(26,52)(27,54) );
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)], [(1,13,34),(2,14,35),(3,15,36),(4,26,12),(5,27,10),(6,25,11),(7,19,41),(8,20,42),(9,21,40),(16,50,47),(17,51,48),(18,49,46),(22,29,45),(23,30,43),(24,28,44),(31,38,52),(32,39,53),(33,37,54)], [(1,25,20),(2,26,21),(3,27,19),(4,9,35),(5,7,36),(6,8,34),(10,41,15),(11,42,13),(12,40,14),(16,37,28),(17,38,29),(18,39,30),(22,48,31),(23,46,32),(24,47,33),(43,49,53),(44,50,54),(45,51,52)], [(1,15,35),(2,13,36),(3,14,34),(4,25,10),(5,26,11),(6,27,12),(7,21,42),(8,19,40),(9,20,41),(16,48,49),(17,46,50),(18,47,51),(22,43,28),(23,44,29),(24,45,30),(31,53,37),(32,54,38),(33,52,39)], [(1,7,6),(2,8,4),(3,9,5),(10,36,40),(11,34,41),(12,35,42),(13,19,25),(14,20,26),(15,21,27),(16,23,39),(17,24,37),(18,22,38),(28,54,51),(29,52,49),(30,53,50),(31,46,45),(32,47,43),(33,48,44)], [(1,43),(2,45),(3,44),(4,31),(5,33),(6,32),(7,47),(8,46),(9,48),(10,37),(11,39),(12,38),(13,30),(14,29),(15,28),(16,41),(17,40),(18,42),(19,50),(20,49),(21,51),(22,35),(23,34),(24,36),(25,53),(26,52),(27,54)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 3O | ··· | 3AX | 6A | 6B |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | 6 |
| size | 1 | 81 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 81 | 81 |
54 irreducible representations
| dim | 1 | 1 | 2 | 2 | 3 | 6 |
| type | + | + | + | + | ||
| image | C1 | C2 | S3 | S3 | He3⋊C2 | He3⋊5S3 |
| kernel | C34⋊13S3 | C32×He3 | C3×He3 | C34 | C32 | C3 |
| # reps | 1 | 1 | 36 | 4 | 4 | 8 |
Matrix representation of C34⋊13S3 ►in GL7(𝔽7)
| 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 |
G:=sub<GL(7,GF(7))| [0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,6,0] >;
C34⋊13S3 in GAP, Magma, Sage, TeX
C_3^4\rtimes_{13}S_3 % in TeX
G:=Group("C3^4:13S3"); // GroupNames label
G:=SmallGroup(486,248);
// by ID
G=gap.SmallGroup(486,248);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,49,218,867,3244,382]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=c*d=d*c,f*c*f=c^-1,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations