direct product, non-abelian, supersoluble, monomial
Aliases: C4xHe3:C2, (C3xC12):4S3, He3:5(C2xC4), (C4xHe3):4C2, C32:4(C4xS3), (C3xC6).17D6, He3:3C4:4C2, C12.11(C3:S3), (C2xHe3).12C22, C3.2(C4xC3:S3), C6.19(C2xC3:S3), C2.1(C2xHe3:C2), (C2xHe3:C2).3C2, SmallGroup(216,67)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C4xHe3:C2 |
Generators and relations for C4xHe3:C2
G = < a,b,c,d,e | a4=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ce=ec, ede=d-1 >
Subgroups: 272 in 88 conjugacy classes, 26 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2xC4, C32, Dic3, C12, C12, D6, C2xC6, C3xS3, C3xC6, C4xS3, C2xC12, He3, C3xDic3, C3xC12, S3xC6, He3:C2, C2xHe3, S3xC12, He3:3C4, C4xHe3, C2xHe3:C2, C4xHe3:C2
Quotients: C1, C2, C4, C22, S3, C2xC4, D6, C3:S3, C4xS3, C2xC3:S3, He3:C2, C4xC3:S3, C2xHe3:C2, C4xHe3:C2
►On 36 points
(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)
(1 16 7)(2 13 8)(3 14 5)(4 15 6)(9 29 18)(10 30 19)(11 31 20)(12 32 17)(21 27 33)(22 28 34)(23 25 35)(24 26 36)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(1 33 27)(2 34 28)(3 35 25)(4 36 26)(5 14 31)(6 15 32)(7 16 29)(8 13 30)(9 18 21)(10 19 22)(11 20 23)(12 17 24)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 20)(10 17)(11 18)(12 19)(21 23)(22 24)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)
G:=sub<Sym(36)| (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), (1,16,7)(2,13,8)(3,14,5)(4,15,6)(9,29,18)(10,30,19)(11,31,20)(12,32,17)(21,27,33)(22,28,34)(23,25,35)(24,26,36), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,33,27)(2,34,28)(3,35,25)(4,36,26)(5,14,31)(6,15,32)(7,16,29)(8,13,30)(9,18,21)(10,19,22)(11,20,23)(12,17,24), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,20)(10,17)(11,18)(12,19)(21,23)(22,24)(25,33)(26,34)(27,35)(28,36)(29,31)(30,32)>;
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), (1,16,7)(2,13,8)(3,14,5)(4,15,6)(9,29,18)(10,30,19)(11,31,20)(12,32,17)(21,27,33)(22,28,34)(23,25,35)(24,26,36), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,33,27)(2,34,28)(3,35,25)(4,36,26)(5,14,31)(6,15,32)(7,16,29)(8,13,30)(9,18,21)(10,19,22)(11,20,23)(12,17,24), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,20)(10,17)(11,18)(12,19)(21,23)(22,24)(25,33)(26,34)(27,35)(28,36)(29,31)(30,32) );
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)], [(1,16,7),(2,13,8),(3,14,5),(4,15,6),(9,29,18),(10,30,19),(11,31,20),(12,32,17),(21,27,33),(22,28,34),(23,25,35),(24,26,36)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,11),(6,36,12),(7,33,9),(8,34,10),(13,28,19),(14,25,20),(15,26,17),(16,27,18)], [(1,33,27),(2,34,28),(3,35,25),(4,36,26),(5,14,31),(6,15,32),(7,16,29),(8,13,30),(9,18,21),(10,19,22),(11,20,23),(12,17,24)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,20),(10,17),(11,18),(12,19),(21,23),(22,24),(25,33),(26,34),(27,35),(28,36),(29,31),(30,32)]])
C4xHe3:C2 is a maximal subgroup of
C12.89S32 He3:3M4(2) He3:6M4(2) He3:2(C2xC8) He3:1M4(2) C4:(He3:C4) C12.84S32 C12.91S32 C12.85S32 C12.86S32 C62.47D6 C62.16D6 He3:5D4:C2
C4xHe3:C2 is a maximal quotient of
He3:6M4(2) C62.29D6 C62.31D6
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | 12N | 12O | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 9 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 9 | 9 | 9 | 9 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4xS3 | He3:C2 | C2xHe3:C2 | C4xHe3:C2 |
| kernel | C4xHe3:C2 | He3:3C4 | C4xHe3 | C2xHe3:C2 | He3:C2 | C3xC12 | C3xC6 | C32 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of C4xHe3:C2 ►in GL3(F13) generated by
| 5 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 6 | 7 | 0 |
| 1 | 7 | 12 |
| 7 | 4 | 0 |
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 2 | 1 | 5 |
| 8 | 11 | 0 |
| 8 | 12 | 0 |
| 1 | 6 | 10 |
| 0 | 12 | 1 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(13))| [5,0,0,0,5,0,0,0,5],[6,1,7,7,7,4,0,12,0],[3,0,0,0,3,0,0,0,3],[2,8,8,1,11,12,5,0,0],[1,0,0,6,12,0,10,1,1] >;
C4xHe3:C2 in GAP, Magma, Sage, TeX
C_4\times {\rm He}_3\rtimes C_2 % in TeX
G:=Group("C4xHe3:C2"); // GroupNames label
G:=SmallGroup(216,67);
// by ID
G=gap.SmallGroup(216,67);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,387,1444,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^3=e^2=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=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations