direct product, metabelian, soluble, monomial
Aliases: C2×C62.C4, (C3×C6)⋊4M4(2), (C2×C62).7C4, C62.20(C2×C4), C32⋊9(C2×M4(2)), C32⋊2C8⋊9C22, C23.3(C32⋊C4), C3⋊Dic3.35C23, (C2×C32⋊2C8)⋊9C2, (C2×C3⋊Dic3).24C4, C3⋊Dic3.55(C2×C4), (C3×C6).35(C22×C4), C22.20(C2×C32⋊C4), C2.12(C22×C32⋊C4), (C22×C3⋊Dic3).14C2, (C2×C3⋊Dic3).176C22, SmallGroup(288,940)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C62.C4
G = < a,b,c,d | a2=b6=c6=1, d4=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >
Subgroups: 448 in 122 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C6 [×14], C8 [×4], C2×C4 [×6], C23, C32, Dic3 [×8], C2×C6 [×14], C2×C8 [×2], M4(2) [×4], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×12], C22×C6 [×2], C2×M4(2), C3⋊Dic3 [×2], C3⋊Dic3 [×2], C62, C62 [×2], C62 [×2], C22×Dic3 [×2], C32⋊2C8 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×4], C2×C62, C2×C32⋊2C8 [×2], C62.C4 [×4], C22×C3⋊Dic3, C2×C62.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, C2×M4(2), C32⋊C4, C2×C32⋊C4 [×3], C62.C4 [×2], C22×C32⋊C4, C2×C62.C4
►On 48 points
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)
(1 46 40)(2 37 47 6 33 43)(3 34 48)(4 45 35 8 41 39)(5 42 36)(7 38 44)(9 24 26)(10 31 17 14 27 21)(11 28 18)(12 23 29 16 19 25)(13 20 30)(15 32 22)
(1 5)(2 43 33 6 47 37)(3 7)(4 39 41 8 35 45)(9 13)(10 21 27 14 17 31)(11 15)(12 25 19 16 29 23)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)
(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)
G:=sub<Sym(48)| (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,46,40)(2,37,47,6,33,43)(3,34,48)(4,45,35,8,41,39)(5,42,36)(7,38,44)(9,24,26)(10,31,17,14,27,21)(11,28,18)(12,23,29,16,19,25)(13,20,30)(15,32,22), (1,5)(2,43,33,6,47,37)(3,7)(4,39,41,8,35,45)(9,13)(10,21,27,14,17,31)(11,15)(12,25,19,16,29,23)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (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)>;
G:=Group( (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (1,46,40)(2,37,47,6,33,43)(3,34,48)(4,45,35,8,41,39)(5,42,36)(7,38,44)(9,24,26)(10,31,17,14,27,21)(11,28,18)(12,23,29,16,19,25)(13,20,30)(15,32,22), (1,5)(2,43,33,6,47,37)(3,7)(4,39,41,8,35,45)(9,13)(10,21,27,14,17,31)(11,15)(12,25,19,16,29,23)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (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) );
G=PermutationGroup([(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44)], [(1,46,40),(2,37,47,6,33,43),(3,34,48),(4,45,35,8,41,39),(5,42,36),(7,38,44),(9,24,26),(10,31,17,14,27,21),(11,28,18),(12,23,29,16,19,25),(13,20,30),(15,32,22)], [(1,5),(2,43,33,6,47,37),(3,7),(4,39,41,8,35,45),(9,13),(10,21,27,14,17,31),(11,15),(12,25,19,16,29,23),(18,22),(20,24),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48)], [(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)])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6N | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 9 | 9 | 9 | 9 | 18 | 18 | 4 | ··· | 4 | 18 | ··· | 18 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | C32⋊C4 | C2×C32⋊C4 | C62.C4 |
| kernel | C2×C62.C4 | C2×C32⋊2C8 | C62.C4 | C22×C3⋊Dic3 | C2×C3⋊Dic3 | C2×C62 | C3×C6 | C23 | C22 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 4 | 2 | 6 | 8 |
Matrix representation of C2×C62.C4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 68 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 47 | 48 | 0 | 0 | 0 | 0 |
| 61 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 19 | 68 | 0 | 0 |
| 0 | 0 | 14 | 54 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,68,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[47,61,0,0,0,0,48,26,0,0,0,0,0,0,0,0,19,14,0,0,0,0,68,54,0,0,72,0,0,0,0,0,0,72,0,0] >;
C2×C62.C4 in GAP, Magma, Sage, TeX
C_2\times C_6^2.C_4
% in TeX
G:=Group("C2xC6^2.C4"); // GroupNames label
G:=SmallGroup(288,940);
// by ID
G=gap.SmallGroup(288,940);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,422,80,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^6=1,d^4=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations