direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×Q8○M4(2), C24.81C23, C12.94C24, C8○D4⋊12C6, C4○D4.5C12, (C6×D4).22C4, D4.9(C2×C12), (C6×Q8).18C4, (C2×C24)⋊39C22, (C2×D4).10C12, C6.64(C23×C4), C8.14(C22×C6), C4.18(C23×C6), (C2×Q8).11C12, Q8.15(C2×C12), (C6×M4(2))⋊34C2, (C2×M4(2))⋊16C6, M4(2)⋊12(C2×C6), C4.23(C22×C12), C2.12(C23×C12), C23.13(C2×C12), C12.168(C22×C4), (C2×C12).970C23, C22.5(C22×C12), (C3×M4(2))⋊41C22, (C22×C12).464C22, (C2×C8)⋊9(C2×C6), (C3×C8○D4)⋊17C2, (C3×C4○D4).6C4, (C2×C4).32(C2×C12), C4○D4.22(C2×C6), (C2×C4○D4).17C6, (C6×C4○D4).25C2, (C3×D4).31(C2×C4), (C3×Q8).34(C2×C4), (C2×C12).204(C2×C4), (C2×C6).36(C22×C4), (C22×C4).80(C2×C6), (C22×C6).25(C2×C4), (C2×C4).140(C22×C6), (C3×C4○D4).60C22, SmallGroup(192,1457)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 290 in 258 conjugacy classes, 238 normal (18 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C6, C6 [×7], C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C12 [×2], C12 [×6], C2×C6, C2×C6 [×6], C2×C6 [×3], C2×C8 [×12], M4(2) [×16], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C24 [×8], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C2×C24 [×12], C3×M4(2) [×16], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], Q8○M4(2), C6×M4(2) [×6], C3×C8○D4 [×8], C6×C4○D4, C3×Q8○M4(2)
Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, C2×C12 [×28], C22×C6 [×15], C23×C4, C22×C12 [×14], C23×C6, Q8○M4(2), C23×C12, C3×Q8○M4(2)
Generators and relations
G = < a,b,c,d,e | a3=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►On 48 points
(1 34 47)(2 35 48)(3 36 41)(4 37 42)(5 38 43)(6 39 44)(7 40 45)(8 33 46)(9 24 28)(10 17 29)(11 18 30)(12 19 31)(13 20 32)(14 21 25)(15 22 26)(16 23 27)
(1 24 5 20)(2 17 6 21)(3 18 7 22)(4 19 8 23)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(25 35 29 39)(26 36 30 40)(27 37 31 33)(28 38 32 34)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 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)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)
G:=sub<Sym(48)| (1,34,47)(2,35,48)(3,36,41)(4,37,42)(5,38,43)(6,39,44)(7,40,45)(8,33,46)(9,24,28)(10,17,29)(11,18,30)(12,19,31)(13,20,32)(14,21,25)(15,22,26)(16,23,27), (1,24,5,20)(2,17,6,21)(3,18,7,22)(4,19,8,23)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(25,35,29,39)(26,36,30,40)(27,37,31,33)(28,38,32,34), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48), (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)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)>;
G:=Group( (1,34,47)(2,35,48)(3,36,41)(4,37,42)(5,38,43)(6,39,44)(7,40,45)(8,33,46)(9,24,28)(10,17,29)(11,18,30)(12,19,31)(13,20,32)(14,21,25)(15,22,26)(16,23,27), (1,24,5,20)(2,17,6,21)(3,18,7,22)(4,19,8,23)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(25,35,29,39)(26,36,30,40)(27,37,31,33)(28,38,32,34), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48), (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)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48) );
G=PermutationGroup([(1,34,47),(2,35,48),(3,36,41),(4,37,42),(5,38,43),(6,39,44),(7,40,45),(8,33,46),(9,24,28),(10,17,29),(11,18,30),(12,19,31),(13,20,32),(14,21,25),(15,22,26),(16,23,27)], [(1,24,5,20),(2,17,6,21),(3,18,7,22),(4,19,8,23),(9,43,13,47),(10,44,14,48),(11,45,15,41),(12,46,16,42),(25,35,29,39),(26,36,30,40),(27,37,31,33),(28,38,32,34)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48)], [(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),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 46 | 0 | 0 | 0 |
| 19 | 27 | 26 | 47 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 27 | 0 |
| 27 | 46 | 47 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 27 |
| 26 | 0 | 38 | 35 |
| 0 | 0 | 1 | 0 |
| 0 | 27 | 0 | 0 |
| 19 | 27 | 26 | 47 |
| 1 | 0 | 45 | 28 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[46,19,0,0,0,27,0,0,0,26,0,27,0,47,27,0],[27,0,0,0,46,46,0,0,47,0,46,0,0,0,0,27],[26,0,0,19,0,0,27,27,38,1,0,26,35,0,0,47],[1,0,0,0,0,1,0,0,45,0,72,0,28,0,0,72] >;
102 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2H | 3A | 3B | 4A | 4B | 4C | ··· | 4I | 6A | 6B | 6C | ··· | 6P | 8A | ··· | 8P | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | Q8○M4(2) | C3×Q8○M4(2) |
| kernel | C3×Q8○M4(2) | C6×M4(2) | C3×C8○D4 | C6×C4○D4 | Q8○M4(2) | C6×D4 | C6×Q8 | C3×C4○D4 | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C3 | C1 |
| # reps | 1 | 6 | 8 | 1 | 2 | 6 | 2 | 8 | 12 | 16 | 2 | 12 | 4 | 16 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_3\times Q_8\circ M_{4(2)} % in TeX
G:=Group("C3xQ8oM4(2)"); // GroupNames label
G:=SmallGroup(192,1457);
// by ID
G=gap.SmallGroup(192,1457);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,1059,2915,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations