direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C4○D4, C6.18C24, C12.55C23, C4○(C6×D4), C4○(C6×Q8), C12○(C2×D4), C12○(C6×D4), D4○(C2×C12), C12○(C2×Q8), C12○(C6×Q8), Q8○(C2×C12), (C2×D4)⋊7C6, D4⋊3(C2×C6), (C2×Q8)⋊8C6, Q8⋊4(C2×C6), C12○(C4○D4), (C6×D4)⋊16C2, (C22×C4)⋊8C6, (C6×Q8)⋊13C2, C4.8(C22×C6), C2.3(C23×C6), (C2×C6).6C23, (C22×C12)⋊13C2, (C2×C12)⋊16C22, (C3×D4)⋊12C22, C23.14(C2×C6), (C3×Q8)⋊11C22, C22.1(C22×C6), (C22×C6).30C22, C4○(C3×C4○D4), (C2×C4)○(C3×D4), (C2×C4)○(C3×Q8), (C2×C4)○(C6×Q8), (C2×C4)⋊5(C2×C6), C12○(C3×C4○D4), (C2×C12)○(C3×D4), (C2×C12)○(C3×Q8), (C2×Q8)○(C2×C12), (C2×C12)○(C6×Q8), SmallGroup(96,223)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×C4○D4
G = < a,b,c,d | a6=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4○D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C6×C4○D4
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C24, C22×C6, C2×C4○D4, C3×C4○D4, C23×C6, C6×C4○D4
►On 48 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)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 29 18 19)(2 30 13 20)(3 25 14 21)(4 26 15 22)(5 27 16 23)(6 28 17 24)(7 33 48 37)(8 34 43 38)(9 35 44 39)(10 36 45 40)(11 31 46 41)(12 32 47 42)
(1 34 18 38)(2 35 13 39)(3 36 14 40)(4 31 15 41)(5 32 16 42)(6 33 17 37)(7 28 48 24)(8 29 43 19)(9 30 44 20)(10 25 45 21)(11 26 46 22)(12 27 47 23)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)(13 42)(14 37)(15 38)(16 39)(17 40)(18 41)(25 48)(26 43)(27 44)(28 45)(29 46)(30 47)
G:=sub<Sym(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), (1,29,18,19)(2,30,13,20)(3,25,14,21)(4,26,15,22)(5,27,16,23)(6,28,17,24)(7,33,48,37)(8,34,43,38)(9,35,44,39)(10,36,45,40)(11,31,46,41)(12,32,47,42), (1,34,18,38)(2,35,13,39)(3,36,14,40)(4,31,15,41)(5,32,16,42)(6,33,17,37)(7,28,48,24)(8,29,43,19)(9,30,44,20)(10,25,45,21)(11,26,46,22)(12,27,47,23), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(13,42)(14,37)(15,38)(16,39)(17,40)(18,41)(25,48)(26,43)(27,44)(28,45)(29,46)(30,47)>;
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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,29,18,19)(2,30,13,20)(3,25,14,21)(4,26,15,22)(5,27,16,23)(6,28,17,24)(7,33,48,37)(8,34,43,38)(9,35,44,39)(10,36,45,40)(11,31,46,41)(12,32,47,42), (1,34,18,38)(2,35,13,39)(3,36,14,40)(4,31,15,41)(5,32,16,42)(6,33,17,37)(7,28,48,24)(8,29,43,19)(9,30,44,20)(10,25,45,21)(11,26,46,22)(12,27,47,23), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,21)(8,22)(9,23)(10,24)(11,19)(12,20)(13,42)(14,37)(15,38)(16,39)(17,40)(18,41)(25,48)(26,43)(27,44)(28,45)(29,46)(30,47) );
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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,29,18,19),(2,30,13,20),(3,25,14,21),(4,26,15,22),(5,27,16,23),(6,28,17,24),(7,33,48,37),(8,34,43,38),(9,35,44,39),(10,36,45,40),(11,31,46,41),(12,32,47,42)], [(1,34,18,38),(2,35,13,39),(3,36,14,40),(4,31,15,41),(5,32,16,42),(6,33,17,37),(7,28,48,24),(8,29,43,19),(9,30,44,20),(10,25,45,21),(11,26,46,22),(12,27,47,23)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,21),(8,22),(9,23),(10,24),(11,19),(12,20),(13,42),(14,37),(15,38),(16,39),(17,40),(18,41),(25,48),(26,43),(27,44),(28,45),(29,46),(30,47)]])
C6×C4○D4 is a maximal subgroup of
C4○D4⋊3Dic3 C4○D4⋊4Dic3 (C6×D4).11C4 (C6×D4)⋊9C4 (C6×D4).16C4 (C3×D4)⋊14D4 (C3×D4).32D4 (C6×D4)⋊10C4 C12.76C24 C12.C24 C6.1042- 1+4 C6.1052- 1+4 C6.1442+ 1+4 (C2×D4)⋊43D6 C6.1452+ 1+4 C6.1462+ 1+4 C6.1072- 1+4 (C2×C12)⋊17D4 C6.1082- 1+4 C6.1482+ 1+4 C6.C25
C6×C4○D4 is a maximal quotient of
D4×C2×C12 Q8×C2×C12
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6F | 6G | ··· | 6R | 12A | ··· | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C4○D4 | C3×C4○D4 |
| kernel | C6×C4○D4 | C22×C12 | C6×D4 | C6×Q8 | C3×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C6 | C2 |
| # reps | 1 | 3 | 3 | 1 | 8 | 2 | 6 | 6 | 2 | 16 | 4 | 8 |
Matrix representation of C6×C4○D4 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,3,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,1,0,0,0,0,1,1,0,0,11,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,11,12] >;
C6×C4○D4 in GAP, Magma, Sage, TeX
C_6\times C_4\circ D_4
% in TeX
G:=Group("C6xC4oD4"); // GroupNames label
G:=SmallGroup(96,223);
// by ID
G=gap.SmallGroup(96,223);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-2,601,230]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations