direct product, non-abelian, soluble
Aliases: C3×He3⋊C4, He3⋊2C12, (C3×He3)⋊1C4, He3⋊C2.2C6, C32.3(C32⋊C4), C3.2(C3×C32⋊C4), (C3×He3⋊C2).1C2, SmallGroup(324,110)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C3×He3⋊C4 |
Generators and relations for C3×He3⋊C4
G = < a,b,c,d,e | a3=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=bcd, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 288 in 54 conjugacy classes, 12 normal (9 characteristic)
C1, C2, C3, C3, C3, C4, S3, C6, C32, C32, C12, C3×S3, C3×C6, He3, He3, C33, C3×C12, He3⋊C2, S3×C32, C3×He3, He3⋊C4, C3×He3⋊C2, C3×He3⋊C4
Quotients: C1, C2, C3, C4, C6, C12, C32⋊C4, He3⋊C4, C3×C32⋊C4, C3×He3⋊C4
►On 54 points
Generators in S54
(1 5 9)(2 6 10)(3 12 8)(4 11 7)(13 17 15)(14 18 16)(19 50 33)(20 47 34)(21 48 31)(22 49 32)(23 52 38)(24 53 35)(25 54 36)(26 51 37)(27 45 40)(28 46 41)(29 43 42)(30 44 39)
(1 23 49)(2 48 41)(3 43 36)(4 35 19)(5 52 32)(6 31 28)(7 53 33)(8 29 54)(9 38 22)(10 21 46)(11 24 50)(12 42 25)(13 44 37)(14 20 45)(15 30 51)(16 34 27)(17 39 26)(18 47 40)
(1 12 18)(2 11 17)(3 14 9)(4 13 10)(5 8 16)(6 7 15)(19 37 46)(20 38 43)(21 35 44)(22 36 45)(23 42 47)(24 39 48)(25 40 49)(26 41 50)(27 32 54)(28 33 51)(29 34 52)(30 31 53)
(2 48 50)(4 35 37)(6 31 33)(7 53 51)(10 21 19)(11 24 26)(13 44 46)(15 30 28)(17 39 41)(20 38 43)(22 45 36)(23 42 47)(25 49 40)(27 54 32)(29 34 52)
(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)
G:=sub<Sym(54)| (1,5,9)(2,6,10)(3,12,8)(4,11,7)(13,17,15)(14,18,16)(19,50,33)(20,47,34)(21,48,31)(22,49,32)(23,52,38)(24,53,35)(25,54,36)(26,51,37)(27,45,40)(28,46,41)(29,43,42)(30,44,39), (1,23,49)(2,48,41)(3,43,36)(4,35,19)(5,52,32)(6,31,28)(7,53,33)(8,29,54)(9,38,22)(10,21,46)(11,24,50)(12,42,25)(13,44,37)(14,20,45)(15,30,51)(16,34,27)(17,39,26)(18,47,40), (1,12,18)(2,11,17)(3,14,9)(4,13,10)(5,8,16)(6,7,15)(19,37,46)(20,38,43)(21,35,44)(22,36,45)(23,42,47)(24,39,48)(25,40,49)(26,41,50)(27,32,54)(28,33,51)(29,34,52)(30,31,53), (2,48,50)(4,35,37)(6,31,33)(7,53,51)(10,21,19)(11,24,26)(13,44,46)(15,30,28)(17,39,41)(20,38,43)(22,45,36)(23,42,47)(25,49,40)(27,54,32)(29,34,52), (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)>;
G:=Group( (1,5,9)(2,6,10)(3,12,8)(4,11,7)(13,17,15)(14,18,16)(19,50,33)(20,47,34)(21,48,31)(22,49,32)(23,52,38)(24,53,35)(25,54,36)(26,51,37)(27,45,40)(28,46,41)(29,43,42)(30,44,39), (1,23,49)(2,48,41)(3,43,36)(4,35,19)(5,52,32)(6,31,28)(7,53,33)(8,29,54)(9,38,22)(10,21,46)(11,24,50)(12,42,25)(13,44,37)(14,20,45)(15,30,51)(16,34,27)(17,39,26)(18,47,40), (1,12,18)(2,11,17)(3,14,9)(4,13,10)(5,8,16)(6,7,15)(19,37,46)(20,38,43)(21,35,44)(22,36,45)(23,42,47)(24,39,48)(25,40,49)(26,41,50)(27,32,54)(28,33,51)(29,34,52)(30,31,53), (2,48,50)(4,35,37)(6,31,33)(7,53,51)(10,21,19)(11,24,26)(13,44,46)(15,30,28)(17,39,41)(20,38,43)(22,45,36)(23,42,47)(25,49,40)(27,54,32)(29,34,52), (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) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,12,8),(4,11,7),(13,17,15),(14,18,16),(19,50,33),(20,47,34),(21,48,31),(22,49,32),(23,52,38),(24,53,35),(25,54,36),(26,51,37),(27,45,40),(28,46,41),(29,43,42),(30,44,39)], [(1,23,49),(2,48,41),(3,43,36),(4,35,19),(5,52,32),(6,31,28),(7,53,33),(8,29,54),(9,38,22),(10,21,46),(11,24,50),(12,42,25),(13,44,37),(14,20,45),(15,30,51),(16,34,27),(17,39,26),(18,47,40)], [(1,12,18),(2,11,17),(3,14,9),(4,13,10),(5,8,16),(6,7,15),(19,37,46),(20,38,43),(21,35,44),(22,36,45),(23,42,47),(24,39,48),(25,40,49),(26,41,50),(27,32,54),(28,33,51),(29,34,52),(30,31,53)], [(2,48,50),(4,35,37),(6,31,33),(7,53,51),(10,21,19),(11,24,26),(13,44,46),(15,30,28),(17,39,41),(20,38,43),(22,45,36),(23,42,47),(25,49,40),(27,54,32),(29,34,52)], [(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)]])
42 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 4A | 4B | 6A | ··· | 6H | 12A | ··· | 12P |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 9 | 1 | ··· | 1 | 12 | ··· | 12 | 9 | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | He3⋊C4 | C32⋊C4 | C3×C32⋊C4 |
| kernel | C3×He3⋊C4 | C3×He3⋊C2 | He3⋊C4 | C3×He3 | He3⋊C2 | He3 | C3 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 24 | 2 | 4 |
Matrix representation of C3×He3⋊C4 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 9 |
| 8 | 0 | 0 | 0 |
| 0 | 10 | 10 | 4 |
| 0 | 10 | 4 | 10 |
| 0 | 12 | 4 | 4 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,0,3,0,0,0,0,3,0,3,0,0],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[8,0,0,0,0,10,10,12,0,10,4,4,0,4,10,4] >;
C3×He3⋊C4 in GAP, Magma, Sage, TeX
C_3\times {\rm He}_3\rtimes C_4 % in TeX
G:=Group("C3xHe3:C4"); // GroupNames label
G:=SmallGroup(324,110);
// by ID
G=gap.SmallGroup(324,110);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,3,-3,36,2019,111,2884,916,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b*c*d,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations