direct product, non-abelian, supersoluble, monomial
Aliases: C2×He3.4C6, (C3×C18)⋊8S3, (C3×C9)⋊12D6, C18.6(C3⋊S3), He3.4(C2×C6), C32.4(S3×C6), C9○He3⋊3C22, (C2×He3).13C6, He3⋊C2.4C6, C9.2(C2×C3⋊S3), C3.7(C6×C3⋊S3), C6.15(C3×C3⋊S3), (C3×C6).12(C3×S3), (C2×C9○He3)⋊2C2, (C2×He3⋊C2).2C3, SmallGroup(324,148)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C2×He3.4C6 |
Generators and relations for C2×He3.4C6
G = < a,b,c,d,e | a2=b3=c3=d3=1, e6=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 277 in 101 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C9, C32, D6, C2×C6, C18, C18, C3×S3, C3×C6, C3×C9, He3, 3- 1+2, C2×C18, S3×C6, S3×C9, He3⋊C2, C3×C18, C2×He3, C2×3- 1+2, C9○He3, S3×C18, C2×He3⋊C2, He3.4C6, C2×C9○He3, C2×He3.4C6
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, C3⋊S3, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C6×C3⋊S3, He3.4C6, C2×He3.4C6
►On 54 points
Generators in S54
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 37)(35 38)(36 39)
(1 52 22)(2 23 53)(3 54 24)(4 25 37)(5 38 26)(6 27 39)(7 40 28)(8 29 41)(9 42 30)(10 31 43)(11 44 32)(12 33 45)(13 46 34)(14 35 47)(15 48 36)(16 19 49)(17 50 20)(18 21 51)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)
(19 25 31)(20 32 26)(21 27 33)(22 34 28)(23 29 35)(24 36 30)(37 49 43)(38 44 50)(39 51 45)(40 46 52)(41 53 47)(42 48 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)
G:=sub<Sym(54)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,37)(35,38)(36,39), (1,52,22)(2,23,53)(3,54,24)(4,25,37)(5,38,26)(6,27,39)(7,40,28)(8,29,41)(9,42,30)(10,31,43)(11,44,32)(12,33,45)(13,46,34)(14,35,47)(15,48,36)(16,19,49)(17,50,20)(18,21,51), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,43,49)(38,44,50)(39,45,51)(40,46,52)(41,47,53)(42,48,54), (19,25,31)(20,32,26)(21,27,33)(22,34,28)(23,29,35)(24,36,30)(37,49,43)(38,44,50)(39,51,45)(40,46,52)(41,53,47)(42,48,54), (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)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,37)(35,38)(36,39), (1,52,22)(2,23,53)(3,54,24)(4,25,37)(5,38,26)(6,27,39)(7,40,28)(8,29,41)(9,42,30)(10,31,43)(11,44,32)(12,33,45)(13,46,34)(14,35,47)(15,48,36)(16,19,49)(17,50,20)(18,21,51), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,43,49)(38,44,50)(39,45,51)(40,46,52)(41,47,53)(42,48,54), (19,25,31)(20,32,26)(21,27,33)(22,34,28)(23,29,35)(24,36,30)(37,49,43)(38,44,50)(39,51,45)(40,46,52)(41,53,47)(42,48,54), (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) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,37),(35,38),(36,39)], [(1,52,22),(2,23,53),(3,54,24),(4,25,37),(5,38,26),(6,27,39),(7,40,28),(8,29,41),(9,42,30),(10,31,43),(11,44,32),(12,33,45),(13,46,34),(14,35,47),(15,48,36),(16,19,49),(17,50,20),(18,21,51)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(37,43,49),(38,44,50),(39,45,51),(40,46,52),(41,47,53),(42,48,54)], [(19,25,31),(20,32,26),(21,27,33),(22,34,28),(23,29,35),(24,36,30),(37,49,43),(38,44,50),(39,51,45),(40,46,52),(41,53,47),(42,48,54)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 9A | ··· | 9F | 9G | ··· | 9N | 18A | ··· | 18F | 18G | ··· | 18N | 18O | ··· | 18Z |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | He3.4C6 | C2×He3.4C6 |
| kernel | C2×He3.4C6 | He3.4C6 | C2×C9○He3 | C2×He3⋊C2 | He3⋊C2 | C2×He3 | C3×C18 | C3×C9 | C3×C6 | C32 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 12 | 12 |
Matrix representation of C2×He3.4C6 ►in GL3(𝔽19) generated by
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 1 | 10 | 0 |
| 0 | 18 | 1 |
| 0 | 18 | 0 |
| 7 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 1 | 0 | 0 |
| 1 | 11 | 0 |
| 12 | 0 | 7 |
| 3 | 0 | 0 |
| 0 | 0 | 3 |
| 0 | 3 | 0 |
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[1,0,0,10,18,18,0,1,0],[7,0,0,0,7,0,0,0,7],[1,1,12,0,11,0,0,0,7],[3,0,0,0,0,3,0,3,0] >;
C2×He3.4C6 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3._4C_6 % in TeX
G:=Group("C2xHe3.4C6"); // GroupNames label
G:=SmallGroup(324,148);
// by ID
G=gap.SmallGroup(324,148);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500,579,2164,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=1,e^6=c,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^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations