direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4○D8, C12.85C24, C24.48C23, 2+ (1+4)⋊8C6, C4○D8⋊4C6, C8○D4⋊7C6, D8⋊7(C2×C6), (C6×D8)⋊26C2, (C2×D8)⋊12C6, C8⋊C22⋊4C6, Q16⋊7(C2×C6), C4.45(C6×D4), SD16⋊4(C2×C6), D4.11(C3×D4), (C3×D4).45D4, C4.8(C23×C6), Q8.16(C3×D4), (C3×Q8).45D4, C22.7(C6×D4), (C2×C24)⋊23C22, C12.406(C2×D4), (C6×D4)⋊40C22, (C3×D8)⋊21C22, M4(2)⋊6(C2×C6), C8.10(C22×C6), D4.5(C22×C6), Q8.9(C22×C6), (C3×Q16)⋊21C22, (C3×D4).38C23, C6.206(C22×D4), (C3×Q8).39C23, (C2×C12).687C23, (C3×SD16)⋊20C22, (C3×2+ (1+4))⋊9C2, (C3×M4(2))⋊27C22, (C2×C8)⋊4(C2×C6), C2.30(D4×C2×C6), C4○D4⋊2(C2×C6), (C3×C8○D4)⋊8C2, (C2×D4)⋊7(C2×C6), (C3×C4○D8)⋊11C2, (C3×C8⋊C22)⋊11C2, (C2×C6).184(C2×D4), (C3×C4○D4)⋊14C22, (C2×C4).48(C22×C6), SmallGroup(192,1465)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 474 in 268 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C6, C6 [×9], C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×9], D4 [×12], Q8, Q8 [×2], C23 [×6], C12, C12 [×3], C12 [×2], C2×C6 [×3], C2×C6 [×12], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×6], C2×D4 [×6], C4○D4, C4○D4 [×6], C4○D4 [×2], C24, C24 [×3], C2×C12 [×3], C2×C12 [×6], C3×D4 [×9], C3×D4 [×12], C3×Q8, C3×Q8 [×2], C22×C6 [×6], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4) [×2], C2×C24 [×3], C3×M4(2) [×3], C3×D8 [×9], C3×SD16 [×6], C3×Q16, C6×D4 [×6], C6×D4 [×6], C3×C4○D4, C3×C4○D4 [×6], C3×C4○D4 [×2], D4○D8, C3×C8○D4, C6×D8 [×3], C3×C4○D8 [×3], C3×C8⋊C22 [×6], C3×2+ (1+4) [×2], C3×D4○D8
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C24, C3×D4 [×4], C22×C6 [×15], C22×D4, C6×D4 [×6], C23×C6, D4○D8, D4×C2×C6, C3×D4○D8
Generators and relations
G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >
►On 48 points
(1 33 19)(2 34 20)(3 35 21)(4 36 22)(5 37 23)(6 38 24)(7 39 17)(8 40 18)(9 41 26)(10 42 27)(11 43 28)(12 44 29)(13 45 30)(14 46 31)(15 47 32)(16 48 25)
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(25 40 29 36)(26 33 30 37)(27 34 31 38)(28 35 32 39)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 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 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 21)(18 20)(22 24)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)
G:=sub<Sym(48)| (1,33,19)(2,34,20)(3,35,21)(4,36,22)(5,37,23)(6,38,24)(7,39,17)(8,40,18)(9,41,26)(10,42,27)(11,43,28)(12,44,29)(13,45,30)(14,46,31)(15,47,32)(16,48,25), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,40,29,36)(26,33,30,37)(27,34,31,38)(28,35,32,39), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48), (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), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,21)(18,20)(22,24)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(42,48)(43,47)(44,46)>;
G:=Group( (1,33,19)(2,34,20)(3,35,21)(4,36,22)(5,37,23)(6,38,24)(7,39,17)(8,40,18)(9,41,26)(10,42,27)(11,43,28)(12,44,29)(13,45,30)(14,46,31)(15,47,32)(16,48,25), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,40,29,36)(26,33,30,37)(27,34,31,38)(28,35,32,39), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48), (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), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,21)(18,20)(22,24)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(42,48)(43,47)(44,46) );
G=PermutationGroup([(1,33,19),(2,34,20),(3,35,21),(4,36,22),(5,37,23),(6,38,24),(7,39,17),(8,40,18),(9,41,26),(10,42,27),(11,43,28),(12,44,29),(13,45,30),(14,46,31),(15,47,32),(16,48,25)], [(1,45,5,41),(2,46,6,42),(3,47,7,43),(4,48,8,44),(9,19,13,23),(10,20,14,24),(11,21,15,17),(12,22,16,18),(25,40,29,36),(26,33,30,37),(27,34,31,38),(28,35,32,39)], [(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32),(41,45),(42,46),(43,47),(44,48)], [(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)], [(2,8),(3,7),(4,6),(10,16),(11,15),(12,14),(17,21),(18,20),(22,24),(25,27),(28,32),(29,31),(34,40),(35,39),(36,38),(42,48),(43,47),(44,46)])
Matrix representation ►G ⊆ GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 6 | 3 | 5 |
| 3 | 6 | 5 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 3 | 2 |
| 6 | 0 | 1 | 4 |
| 0 | 5 | 4 | 5 |
| 0 | 4 | 5 | 5 |
| 0 | 6 | 6 | 5 |
| 5 | 0 | 5 | 1 |
| 1 | 5 | 2 | 1 |
| 1 | 6 | 2 | 5 |
| 5 | 5 | 1 | 1 |
| 0 | 4 | 5 | 0 |
| 4 | 6 | 1 | 0 |
| 4 | 5 | 2 | 0 |
| 6 | 1 | 4 | 6 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,3,1,1,6,6,0,1,3,5,0,3,5,1,1,2],[6,0,0,0,0,5,4,6,1,4,5,6,4,5,5,5],[5,1,1,5,0,5,6,5,5,2,2,1,1,1,5,1],[0,4,4,6,4,6,5,1,5,1,2,4,0,0,0,6] >;
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6T | 8A | 8B | 8C | 8D | 8E | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
66 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 | C3×D4 | C3×D4 | D4○D8 | C3×D4○D8 |
| kernel | C3×D4○D8 | C3×C8○D4 | C6×D8 | C3×C4○D8 | C3×C8⋊C22 | C3×2+ (1+4) | D4○D8 | C8○D4 | C2×D8 | C4○D8 | C8⋊C22 | 2+ (1+4) | C3×D4 | C3×Q8 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 2 | 2 | 6 | 6 | 12 | 4 | 3 | 1 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_3\times D_4\circ D_8
% in TeX
G:=Group("C3xD4oD8"); // GroupNames label
G:=SmallGroup(192,1465);
// by ID
G=gap.SmallGroup(192,1465);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,745,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations