direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C6×SD16, C12.42D4, C24⋊13C22, C12.45C23, C8⋊3(C2×C6), (C2×C8)⋊5C6, Q8⋊2(C2×C6), (C2×Q8)⋊5C6, C4.7(C3×D4), (C2×C24)⋊13C2, (C6×Q8)⋊10C2, (C2×D4).6C6, D4.1(C2×C6), C2.12(C6×D4), (C2×C6).53D4, C6.75(C2×D4), (C6×D4).13C2, C4.2(C22×C6), (C3×Q8)⋊9C22, C22.15(C3×D4), (C3×D4).11C22, (C2×C12).130C22, (C2×C4).26(C2×C6), SmallGroup(96,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×SD16
G = < a,b,c | a6=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C2×SD16, C2×C24, C3×SD16, C6×D4, C6×Q8, C6×SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, SD16, C2×D4, C3×D4, C22×C6, C2×SD16, C3×SD16, C6×D4, C6×SD16
►On 48 points
(1 10 34 21 41 26)(2 11 35 22 42 27)(3 12 36 23 43 28)(4 13 37 24 44 29)(5 14 38 17 45 30)(6 15 39 18 46 31)(7 16 40 19 47 32)(8 9 33 20 48 25)
(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)
(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)(18 20)(19 23)(22 24)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(42 44)(43 47)(46 48)
G:=sub<Sym(48)| (1,10,34,21,41,26)(2,11,35,22,42,27)(3,12,36,23,43,28)(4,13,37,24,44,29)(5,14,38,17,45,30)(6,15,39,18,46,31)(7,16,40,19,47,32)(8,9,33,20,48,25), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(42,44)(43,47)(46,48)>;
G:=Group( (1,10,34,21,41,26)(2,11,35,22,42,27)(3,12,36,23,43,28)(4,13,37,24,44,29)(5,14,38,17,45,30)(6,15,39,18,46,31)(7,16,40,19,47,32)(8,9,33,20,48,25), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(18,20)(19,23)(22,24)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(42,44)(43,47)(46,48) );
G=PermutationGroup([[(1,10,34,21,41,26),(2,11,35,22,42,27),(3,12,36,23,43,28),(4,13,37,24,44,29),(5,14,38,17,45,30),(6,15,39,18,46,31),(7,16,40,19,47,32),(8,9,33,20,48,25)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(18,20),(19,23),(22,24),(25,31),(27,29),(28,32),(33,39),(35,37),(36,40),(42,44),(43,47),(46,48)]])
C6×SD16 is a maximal subgroup of
Dic3⋊3SD16 Dic3⋊5SD16 SD16⋊Dic3 (C3×D4).D4 (C3×Q8).D4 C24.31D4 C24.43D4 D6⋊6SD16 D6⋊8SD16 C24⋊14D4 D12⋊7D4 Dic6.16D4 C24⋊8D4 C24⋊15D4 C24⋊9D4 C24.44D4 SD16⋊13D6
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | SD16 | C3×D4 | C3×D4 | C3×SD16 |
| kernel | C6×SD16 | C2×C24 | C3×SD16 | C6×D4 | C6×Q8 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 8 |
Matrix representation of C6×SD16 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 67 | 6 |
| 0 | 0 | 67 | 67 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,67,67,0,0,6,67],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;
C6×SD16 in GAP, Magma, Sage, TeX
C_6\times {\rm SD}_{16} % in TeX
G:=Group("C6xSD16"); // GroupNames label
G:=SmallGroup(96,180);
// by ID
G=gap.SmallGroup(96,180);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,2164,1090,88]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations