direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3xC8:C22, D8:2C6, C24:6C22, SD16:1C6, C12.63D4, M4(2):1C6, C12.48C23, C8:(C2xC6), C4oD4:4C6, (C2xD4):5C6, D4:2(C2xC6), (C3xD8):6C2, Q8:3(C2xC6), (C6xD4):14C2, C6.78(C2xD4), (C2xC6).24D4, C4.14(C3xD4), C2.15(C6xD4), (C3xSD16):5C2, C4.5(C22xC6), C22.5(C3xD4), (C3xD4):11C22, (C3xM4(2)):3C2, (C3xQ8):10C22, (C2xC12).69C22, (C3xC4oD4):7C2, (C2xC4).10(C2xC6), SmallGroup(96,183)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC8:C22
G = < a,b,c,d | a3=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, D4, Q8, C23, C12, C12, C2xC6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xD4, C3xQ8, C22xC6, C8:C22, C3xM4(2), C3xD8, C3xSD16, C6xD4, C3xC4oD4, C3xC8:C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C3xD4, C22xC6, C8:C22, C6xD4, C3xC8:C22
►On 24 points - transitive group 24T114
(1 15 19)(2 16 20)(3 9 21)(4 10 22)(5 11 23)(6 12 24)(7 13 17)(8 14 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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)
G:=sub<Sym(24)| (1,15,19)(2,16,20)(3,9,21)(4,10,22)(5,11,23)(6,12,24)(7,13,17)(8,14,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)>;
G:=Group( (1,15,19)(2,16,20)(3,9,21)(4,10,22)(5,11,23)(6,12,24)(7,13,17)(8,14,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,21)(18,24)(20,22), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23) );
G=PermutationGroup([[(1,15,19),(2,16,20),(3,9,21),(4,10,22),(5,11,23),(6,12,24),(7,13,17),(8,14,18)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,21),(18,24),(20,22)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23)]])
G:=TransitiveGroup(24,114);
C3xC8:C22 is a maximal subgroup of
D12:18D4 M4(2).D6 M4(2).13D6 D12.38D4 D8:4D6 D8:5D6 D8:6D6
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | ··· | 6J | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3xD4 | C3xD4 | C8:C22 | C3xC8:C22 |
| kernel | C3xC8:C22 | C3xM4(2) | C3xD8 | C3xSD16 | C6xD4 | C3xC4oD4 | C8:C22 | M4(2) | D8 | SD16 | C2xD4 | C4oD4 | C12 | C2xC6 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 2 |
Matrix representation of C3xC8:C22 ►in GL4(F7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 1 | 0 | 4 |
| 4 | 3 | 3 | 0 |
| 2 | 5 | 6 | 4 |
| 3 | 3 | 2 | 1 |
| 6 | 0 | 3 | 2 |
| 0 | 6 | 2 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 6 | 3 | 2 |
| 6 | 0 | 4 | 2 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,4,2,3,1,3,5,3,0,3,6,2,4,0,4,1],[6,0,0,0,0,6,0,0,3,2,1,0,2,2,0,1],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1] >;
C3xC8:C22 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes C_2^2
% in TeX
G:=Group("C3xC8:C2^2"); // GroupNames label
G:=SmallGroup(96,183);
// by ID
G=gap.SmallGroup(96,183);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,938,2164,1090,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations