direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C6.D4, C24.2S3, C23⋊3Dic3, C23.37D6, (C22×C6)⋊4C4, C6.62(C2×D4), (C2×C6).44D4, C6⋊2(C22⋊C4), (C23×C6).2C2, (C2×C6).60C23, C6.28(C22×C4), C22⋊3(C2×Dic3), (C2×Dic3)⋊7C22, (C22×Dic3)⋊7C2, C2.9(C22×Dic3), C22.25(C3⋊D4), (C22×C6).41C22, C22.27(C22×S3), (C2×C6)⋊8(C2×C4), C3⋊3(C2×C22⋊C4), C2.4(C2×C3⋊D4), SmallGroup(96,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6.D4
G = < a,b,c,d | a2=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 242 in 132 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, Dic3, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C2×Dic3, C2×Dic3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C6.D4, C22×Dic3, C23×C6, C2×C6.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C6.D4, C22×Dic3, C2×C3⋊D4, C2×C6.D4
►On 48 points
(1 27)(2 28)(3 29)(4 30)(5 25)(6 26)(7 41)(8 42)(9 37)(10 38)(11 39)(12 40)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(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)
(1 31 14 38)(2 36 15 37)(3 35 16 42)(4 34 17 41)(5 33 18 40)(6 32 13 39)(7 30 43 23)(8 29 44 22)(9 28 45 21)(10 27 46 20)(11 26 47 19)(12 25 48 24)
(1 41 4 38)(2 40 5 37)(3 39 6 42)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 35 16 32)(14 34 17 31)(15 33 18 36)(19 44 22 47)(20 43 23 46)(21 48 24 45)
G:=sub<Sym(48)| (1,27)(2,28)(3,29)(4,30)(5,25)(6,26)(7,41)(8,42)(9,37)(10,38)(11,39)(12,40)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (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), (1,31,14,38)(2,36,15,37)(3,35,16,42)(4,34,17,41)(5,33,18,40)(6,32,13,39)(7,30,43,23)(8,29,44,22)(9,28,45,21)(10,27,46,20)(11,26,47,19)(12,25,48,24), (1,41,4,38)(2,40,5,37)(3,39,6,42)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,35,16,32)(14,34,17,31)(15,33,18,36)(19,44,22,47)(20,43,23,46)(21,48,24,45)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,25)(6,26)(7,41)(8,42)(9,37)(10,38)(11,39)(12,40)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (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), (1,31,14,38)(2,36,15,37)(3,35,16,42)(4,34,17,41)(5,33,18,40)(6,32,13,39)(7,30,43,23)(8,29,44,22)(9,28,45,21)(10,27,46,20)(11,26,47,19)(12,25,48,24), (1,41,4,38)(2,40,5,37)(3,39,6,42)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,35,16,32)(14,34,17,31)(15,33,18,36)(19,44,22,47)(20,43,23,46)(21,48,24,45) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,25),(6,26),(7,41),(8,42),(9,37),(10,38),(11,39),(12,40),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)], [(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)], [(1,31,14,38),(2,36,15,37),(3,35,16,42),(4,34,17,41),(5,33,18,40),(6,32,13,39),(7,30,43,23),(8,29,44,22),(9,28,45,21),(10,27,46,20),(11,26,47,19),(12,25,48,24)], [(1,41,4,38),(2,40,5,37),(3,39,6,42),(7,30,10,27),(8,29,11,26),(9,28,12,25),(13,35,16,32),(14,34,17,31),(15,33,18,36),(19,44,22,47),(20,43,23,46),(21,48,24,45)]])
C2×C6.D4 is a maximal subgroup of
C24.12D6 C24.13D6 Dic3×C22⋊C4 C24.55D6 C24.56D6 C24.14D6 C24.15D6 C24.57D6 C23⋊2Dic6 C24.17D6 C24.18D6 C24.58D6 C24.19D6 C24.20D6 C24.21D6 C24.59D6 C24.23D6 C24.24D6 C24.25D6 C24.27D6 C24.73D6 C24.74D6 C24.75D6 C24.76D6 C24.29D6 C24.30D6 C24.31D6 C24.32D6 C25.4S3 C23⋊3Dic6 C2×S3×C22⋊C4 C24.35D6 C24.42D6 C24.43D6 C24.44D6 C24.46D6 C2×C4×C3⋊D4 C2×D4×Dic3 C24.49D6 C24⋊12D6 C24.53D6
C2×C6.D4 is a maximal quotient of
C24.6Dic3 C24.74D6 C24.75D6 (C6×D4)⋊6C4 C24.29D6 C24.30D6 (C6×Q8)⋊6C4 (C6×Q8)⋊7C4 C4○D4⋊3Dic3 C4○D4⋊4Dic3 (C6×D4).11C4 (C6×D4)⋊9C4 (C6×D4).16C4 (C6×D4)⋊10C4 C25.4S3
36 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 6A | ··· | 6O |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | C3⋊D4 |
| kernel | C2×C6.D4 | C6.D4 | C22×Dic3 | C23×C6 | C22×C6 | C24 | C2×C6 | C23 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 4 | 4 | 3 | 8 |
Matrix representation of C2×C6.D4 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 9 |
| 8 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,3,0,0,0,0,9],[8,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[5,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;
C2×C6.D4 in GAP, Magma, Sage, TeX
C_2\times C_6.D_4
% in TeX
G:=Group("C2xC6.D4"); // GroupNames label
G:=SmallGroup(96,159);
// by ID
G=gap.SmallGroup(96,159);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,362,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^4=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations