direct product, metabelian, supersoluble, monomial
Aliases: C3xC32:7D4, C33:14D4, C62:11S3, C62:12C6, C6.28(S3xC6), C3:Dic3:6C6, (C3xC62):3C2, (C3xC6).61D6, C32:10(C3xD4), C32:12(C3:D4), (C32xC6).25C22, (C6xC3:S3):6C2, (C2xC3:S3):6C6, (C2xC6):6(C3xS3), C2.5(C6xC3:S3), C3:3(C3xC3:D4), (C2xC6):3(C3:S3), C6.26(C2xC3:S3), C22:3(C3xC3:S3), (C3xC6).33(C2xC6), (C3xC3:Dic3):8C2, SmallGroup(216,144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC32:7D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 360 in 136 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, C3:D4, C3xD4, C33, C3xDic3, C3:Dic3, S3xC6, C2xC3:S3, C62, C62, C62, C3xC3:S3, C32xC6, C32xC6, C3xC3:D4, C32:7D4, C3xC3:Dic3, C6xC3:S3, C3xC62, C3xC32:7D4
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2xC6, C3xS3, C3:S3, C3:D4, C3xD4, S3xC6, C2xC3:S3, C3xC3:S3, C3xC3:D4, C32:7D4, C6xC3:S3, C3xC32:7D4
►On 36 points
(1 27 9)(2 28 10)(3 25 11)(4 26 12)(5 23 20)(6 24 17)(7 21 18)(8 22 19)(13 34 30)(14 35 31)(15 36 32)(16 33 29)
(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)
(1 21 29)(2 30 22)(3 23 31)(4 32 24)(5 35 11)(6 12 36)(7 33 9)(8 10 34)(13 19 28)(14 25 20)(15 17 26)(16 27 18)
(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)
G:=sub<Sym(36)| (1,27,9)(2,28,10)(3,25,11)(4,26,12)(5,23,20)(6,24,17)(7,21,18)(8,22,19)(13,34,30)(14,35,31)(15,36,32)(16,33,29), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,30,22)(3,23,31)(4,32,24)(5,35,11)(6,12,36)(7,33,9)(8,10,34)(13,19,28)(14,25,20)(15,17,26)(16,27,18), (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)>;
G:=Group( (1,27,9)(2,28,10)(3,25,11)(4,26,12)(5,23,20)(6,24,17)(7,21,18)(8,22,19)(13,34,30)(14,35,31)(15,36,32)(16,33,29), (1,16,7)(2,8,13)(3,14,5)(4,6,15)(9,29,18)(10,19,30)(11,31,20)(12,17,32)(21,27,33)(22,34,28)(23,25,35)(24,36,26), (1,21,29)(2,30,22)(3,23,31)(4,32,24)(5,35,11)(6,12,36)(7,33,9)(8,10,34)(13,19,28)(14,25,20)(15,17,26)(16,27,18), (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35) );
G=PermutationGroup([[(1,27,9),(2,28,10),(3,25,11),(4,26,12),(5,23,20),(6,24,17),(7,21,18),(8,22,19),(13,34,30),(14,35,31),(15,36,32),(16,33,29)], [(1,16,7),(2,8,13),(3,14,5),(4,6,15),(9,29,18),(10,19,30),(11,31,20),(12,17,32),(21,27,33),(22,34,28),(23,25,35),(24,36,26)], [(1,21,29),(2,30,22),(3,23,31),(4,32,24),(5,35,11),(6,12,36),(7,33,9),(8,10,34),(13,19,28),(14,25,20),(15,17,26),(16,27,18)], [(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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,18),(19,20),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35)]])
C3xC32:7D4 is a maximal subgroup of
C3xS3xC3:D4 C62.91D6 C62.93D6 C62:23D6 C62.96D6 C62:24D6 C3xD4xC3:S3
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4 | 6A | 6B | 6C | ··· | 6AN | 6AO | 6AP | 12A | 12B |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 |
| size | 1 | 1 | 2 | 18 | 1 | 1 | 2 | ··· | 2 | 18 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | C3xS3 | C3:D4 | C3xD4 | S3xC6 | C3xC3:D4 |
| kernel | C3xC32:7D4 | C3xC3:Dic3 | C6xC3:S3 | C3xC62 | C32:7D4 | C3:Dic3 | C2xC3:S3 | C62 | C62 | C33 | C3xC6 | C2xC6 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 4 | 8 | 8 | 2 | 8 | 16 |
Matrix representation of C3xC32:7D4 ►in GL4(F13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 12 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 4 |
| 0 | 0 | 7 | 5 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 4 |
| 0 | 0 | 7 | 5 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,3,12,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,1,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,8,7,0,0,4,5],[0,1,0,0,1,0,0,0,0,0,8,7,0,0,4,5] >;
C3xC32:7D4 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_7D_4
% in TeX
G:=Group("C3xC3^2:7D4"); // GroupNames label
G:=SmallGroup(216,144);
// by ID
G=gap.SmallGroup(216,144);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,1444,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=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=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations