direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C8○D4, D4○C24, Q8○C24, D4.C12, M4(2)○C24, Q8.2C12, M4(2)⋊5C6, C24.30C22, C12.54C23, C8○(C3×D4), C8○(C3×Q8), (C2×C8)⋊7C6, C4○D4○C24, C24○(C3×D4), C24○(C3×Q8), C8.7(C2×C6), (C2×C24)⋊15C2, C4.5(C2×C12), C4○D4.5C6, (C3×D4).2C4, C8○(C3×M4(2)), (C3×Q8).2C4, C12.32(C2×C4), C24○(C3×M4(2)), C2.7(C22×C12), C4.12(C22×C6), C22.1(C2×C12), C6.35(C22×C4), (C3×M4(2))⋊11C2, (C2×C12).128C22, C8○(C3×C4○D4), C24○(C3×C4○D4), (C2×C6).8(C2×C4), (C2×C4).24(C2×C6), (C3×C4○D4).6C2, SmallGroup(96,178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8○D4
G = < a,b,c,d | a3=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C8, M4(2), C4○D4, C24, C24, C2×C12, C3×D4, C3×Q8, C8○D4, C2×C24, C3×M4(2), C3×C4○D4, C3×C8○D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C2×C12, C22×C6, C8○D4, C22×C12, C3×C8○D4
►On 48 points
(1 33 23)(2 34 24)(3 35 17)(4 36 18)(5 37 19)(6 38 20)(7 39 21)(8 40 22)(9 48 28)(10 41 29)(11 42 30)(12 43 31)(13 44 32)(14 45 25)(15 46 26)(16 47 27)
(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 43 5 47)(2 44 6 48)(3 45 7 41)(4 46 8 42)(9 24 13 20)(10 17 14 21)(11 18 15 22)(12 19 16 23)(25 39 29 35)(26 40 30 36)(27 33 31 37)(28 34 32 38)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)
G:=sub<Sym(48)| (1,33,23)(2,34,24)(3,35,17)(4,36,18)(5,37,19)(6,38,20)(7,39,21)(8,40,22)(9,48,28)(10,41,29)(11,42,30)(12,43,31)(13,44,32)(14,45,25)(15,46,26)(16,47,27), (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), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,24,13,20)(10,17,14,21)(11,18,15,22)(12,19,16,23)(25,39,29,35)(26,40,30,36)(27,33,31,37)(28,34,32,38), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)>;
G:=Group( (1,33,23)(2,34,24)(3,35,17)(4,36,18)(5,37,19)(6,38,20)(7,39,21)(8,40,22)(9,48,28)(10,41,29)(11,42,30)(12,43,31)(13,44,32)(14,45,25)(15,46,26)(16,47,27), (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), (1,43,5,47)(2,44,6,48)(3,45,7,41)(4,46,8,42)(9,24,13,20)(10,17,14,21)(11,18,15,22)(12,19,16,23)(25,39,29,35)(26,40,30,36)(27,33,31,37)(28,34,32,38), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38) );
G=PermutationGroup([[(1,33,23),(2,34,24),(3,35,17),(4,36,18),(5,37,19),(6,38,20),(7,39,21),(8,40,22),(9,48,28),(10,41,29),(11,42,30),(12,43,31),(13,44,32),(14,45,25),(15,46,26),(16,47,27)], [(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)], [(1,43,5,47),(2,44,6,48),(3,45,7,41),(4,46,8,42),(9,24,13,20),(10,17,14,21),(11,18,15,22),(12,19,16,23),(25,39,29,35),(26,40,30,36),(27,33,31,37),(28,34,32,38)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38)]])
C3×C8○D4 is a maximal subgroup of
C24.99D4 C24.78C23 Q8.8D12 Q8.9D12 Q8.10D12 C24.100D4 C24.54D4 M4(2)⋊28D6 D4.11D12 D4.12D12 D4.13D12 Q8.C36
C3×C8○D4 is a maximal quotient of
D4×C24 Q8×C24
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24H | 24I | ··· | 24T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C8○D4 | C3×C8○D4 |
| kernel | C3×C8○D4 | C2×C24 | C3×M4(2) | C3×C4○D4 | C8○D4 | C3×D4 | C3×Q8 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 2 | 6 | 6 | 2 | 12 | 4 | 4 | 8 |
Matrix representation of C3×C8○D4 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 72 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 72 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[1,0,0,0,10,0,0,0,10],[72,0,0,0,0,72,0,1,0],[1,0,0,0,0,1,0,1,0] >;
C3×C8○D4 in GAP, Magma, Sage, TeX
C_3\times C_8\circ D_4
% in TeX
G:=Group("C3xC8oD4"); // GroupNames label
G:=SmallGroup(96,178);
// by ID
G=gap.SmallGroup(96,178);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,476,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations