direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8.26D4, D8⋊4C12, Q16⋊4C12, SD16⋊2C12, C24.105D4, C4≀C2⋊6C6, C8⋊C4⋊3C6, C8○D4⋊11C6, (C3×D8)⋊10C4, C8.6(C2×C12), C4○D8.3C6, C8.25(C3×D4), C4.83(C6×D4), C8.C4⋊4C6, C24.41(C2×C4), (C3×Q16)⋊10C4, (C3×SD16)⋊6C4, D4.4(C2×C12), C2.19(D4×C12), C6.121(C4×D4), Q8.9(C2×C12), C12.488(C2×D4), C42.11(C2×C6), C4.16(C22×C12), (C4×C12).252C22, C12.161(C22×C4), (C2×C12).911C23, (C2×C24).199C22, M4(2).12(C2×C6), (C3×M4(2)).46C22, (C3×C4≀C2)⋊14C2, (C3×C8⋊C4)⋊8C2, (C3×C8○D4)⋊16C2, (C2×C8).54(C2×C6), (C3×C4○D8).8C2, C4○D4.16(C2×C6), (C3×D4).21(C2×C4), (C3×Q8).22(C2×C4), (C3×C8.C4)⋊13C2, C22.2(C3×C4○D4), (C2×C6).50(C4○D4), (C2×C4).86(C22×C6), (C3×C4○D4).54C22, SmallGroup(192,877)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.26D4
G = < a,b,c,d | a3=b8=c4=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b2c-1 >
Subgroups: 154 in 104 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C24, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C4×C12, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C8.26D4, C3×C8⋊C4, C3×C4≀C2, C3×C8.C4, C3×C8○D4, C3×C4○D8, C3×C8.26D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C22×C12, C6×D4, C3×C4○D4, C8.26D4, D4×C12, C3×C8.26D4
►On 48 points
(1 27 23)(2 28 24)(3 29 17)(4 30 18)(5 31 19)(6 32 20)(7 25 21)(8 26 22)(9 36 41)(10 37 42)(11 38 43)(12 39 44)(13 40 45)(14 33 46)(15 34 47)(16 35 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)
(2 6)(4 8)(9 11 13 15)(10 16 14 12)(18 22)(20 24)(26 30)(28 32)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)
(1 47 3 41 5 43 7 45)(2 44 4 46 6 48 8 42)(9 31 11 25 13 27 15 29)(10 28 12 30 14 32 16 26)(17 36 19 38 21 40 23 34)(18 33 20 35 22 37 24 39)
G:=sub<Sym(48)| (1,27,23)(2,28,24)(3,29,17)(4,30,18)(5,31,19)(6,32,20)(7,25,21)(8,26,22)(9,36,41)(10,37,42)(11,38,43)(12,39,44)(13,40,45)(14,33,46)(15,34,47)(16,35,48), (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), (2,6)(4,8)(9,11,13,15)(10,16,14,12)(18,22)(20,24)(26,30)(28,32)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44), (1,47,3,41,5,43,7,45)(2,44,4,46,6,48,8,42)(9,31,11,25,13,27,15,29)(10,28,12,30,14,32,16,26)(17,36,19,38,21,40,23,34)(18,33,20,35,22,37,24,39)>;
G:=Group( (1,27,23)(2,28,24)(3,29,17)(4,30,18)(5,31,19)(6,32,20)(7,25,21)(8,26,22)(9,36,41)(10,37,42)(11,38,43)(12,39,44)(13,40,45)(14,33,46)(15,34,47)(16,35,48), (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), (2,6)(4,8)(9,11,13,15)(10,16,14,12)(18,22)(20,24)(26,30)(28,32)(33,39,37,35)(34,36,38,40)(41,43,45,47)(42,48,46,44), (1,47,3,41,5,43,7,45)(2,44,4,46,6,48,8,42)(9,31,11,25,13,27,15,29)(10,28,12,30,14,32,16,26)(17,36,19,38,21,40,23,34)(18,33,20,35,22,37,24,39) );
G=PermutationGroup([[(1,27,23),(2,28,24),(3,29,17),(4,30,18),(5,31,19),(6,32,20),(7,25,21),(8,26,22),(9,36,41),(10,37,42),(11,38,43),(12,39,44),(13,40,45),(14,33,46),(15,34,47),(16,35,48)], [(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)], [(2,6),(4,8),(9,11,13,15),(10,16,14,12),(18,22),(20,24),(26,30),(28,32),(33,39,37,35),(34,36,38,40),(41,43,45,47),(42,48,46,44)], [(1,47,3,41,5,43,7,45),(2,44,4,46,6,48,8,42),(9,31,11,25,13,27,15,29),(10,28,12,30,14,32,16,26),(17,36,19,38,21,40,23,34),(18,33,20,35,22,37,24,39)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12N | 24A | ··· | 24H | 24I | ··· | 24T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | C8.26D4 | C3×C8.26D4 |
| kernel | C3×C8.26D4 | C3×C8⋊C4 | C3×C4≀C2 | C3×C8.C4 | C3×C8○D4 | C3×C4○D8 | C8.26D4 | C3×D8 | C3×SD16 | C3×Q16 | C8⋊C4 | C4≀C2 | C8.C4 | C8○D4 | C4○D8 | D8 | SD16 | Q16 | C24 | C2×C6 | C8 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×C8.26D4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 27 | 59 |
| 46 | 0 | 27 | 0 |
| 0 | 0 | 27 | 59 |
| 0 | 0 | 54 | 46 |
| 1 | 0 | 0 | 13 |
| 0 | 72 | 0 | 14 |
| 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 27 |
| 0 | 1 | 46 | 13 |
| 0 | 1 | 0 | 59 |
| 1 | 1 | 0 | 72 |
| 0 | 2 | 0 | 72 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,46,0,0,1,0,0,0,27,27,27,54,59,0,59,46],[1,0,0,0,0,72,0,0,0,0,46,0,13,14,27,27],[0,0,1,0,1,1,1,2,46,0,0,0,13,59,72,72] >;
C3×C8.26D4 in GAP, Magma, Sage, TeX
C_3\times C_8._{26}D_4 % in TeX
G:=Group("C3xC8.26D4"); // GroupNames label
G:=SmallGroup(192,877);
// by ID
G=gap.SmallGroup(192,877);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,2102,268,4204,2111,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^2*c^-1>;
// generators/relations