metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.46D6, C6.292+ 1+4, C22≀C2⋊7S3, (C2×D4).86D6, C22⋊C4.2D6, D6⋊3D4⋊14C2, C24⋊4S3⋊8C2, D6⋊C4⋊14C22, (D4×Dic3)⋊13C2, C23.14D6⋊5C2, Dic3⋊4D4⋊4C2, C23.9D6⋊14C2, (C2×C12).31C23, (C2×C6).137C24, C4⋊Dic3⋊27C22, C23.12D6⋊12C2, C2.31(D4⋊6D6), Dic3⋊C4⋊12C22, C3⋊3(C22.32C24), (C2×Dic6)⋊22C22, (C4×Dic3)⋊17C22, (C6×D4).111C22, C23.23D6⋊5C2, C23.8D6⋊12C2, (C23×C6).70C22, C23.11D6⋊14C2, Dic3.D4⋊14C2, C6.D4⋊17C22, (C22×S3).56C23, (C22×C6).182C23, C22.158(S3×C23), C23.187(C22×S3), (C2×Dic3).62C23, C22.10(D4⋊2S3), (C22×Dic3)⋊16C22, (S3×C2×C4)⋊10C22, C6.78(C2×C4○D4), (C3×C22≀C2)⋊8C2, (C2×C6).44(C4○D4), C2.29(C2×D4⋊2S3), (C2×C3⋊D4)⋊10C22, (C2×C4).31(C22×S3), (C2×C6.D4)⋊21C2, (C3×C22⋊C4).3C22, SmallGroup(192,1152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.46D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 656 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C6×D4, C23×C6, C22.32C24, Dic3.D4, C23.8D6, Dic3⋊4D4, C23.9D6, C23.11D6, D4×Dic3, C23.23D6, C23.12D6, D6⋊3D4, C23.14D6, C2×C6.D4, C24⋊4S3, C3×C22≀C2, C24.46D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, D4⋊2S3, S3×C23, C22.32C24, C2×D4⋊2S3, D4⋊6D6, C24.46D6
►On 48 points
(2 19)(4 21)(6 23)(8 13)(10 15)(12 17)(25 43)(26 32)(27 45)(28 34)(29 47)(30 36)(31 37)(33 39)(35 41)(38 44)(40 46)(42 48)
(1 7)(3 9)(5 11)(14 20)(16 22)(18 24)(25 37)(26 44)(27 39)(28 46)(29 41)(30 48)(31 43)(32 38)(33 45)(34 40)(35 47)(36 42)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 13)(9 14)(10 15)(11 16)(12 17)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 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)
(1 41 7 47)(2 46 8 40)(3 39 9 45)(4 44 10 38)(5 37 11 43)(6 42 12 48)(13 28 19 34)(14 33 20 27)(15 26 21 32)(16 31 22 25)(17 36 23 30)(18 29 24 35)
G:=sub<Sym(48)| (2,19)(4,21)(6,23)(8,13)(10,15)(12,17)(25,43)(26,32)(27,45)(28,34)(29,47)(30,36)(31,37)(33,39)(35,41)(38,44)(40,46)(42,48), (1,7)(3,9)(5,11)(14,20)(16,22)(18,24)(25,37)(26,44)(27,39)(28,46)(29,41)(30,48)(31,43)(32,38)(33,45)(34,40)(35,47)(36,42), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(12,17)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,41,7,47)(2,46,8,40)(3,39,9,45)(4,44,10,38)(5,37,11,43)(6,42,12,48)(13,28,19,34)(14,33,20,27)(15,26,21,32)(16,31,22,25)(17,36,23,30)(18,29,24,35)>;
G:=Group( (2,19)(4,21)(6,23)(8,13)(10,15)(12,17)(25,43)(26,32)(27,45)(28,34)(29,47)(30,36)(31,37)(33,39)(35,41)(38,44)(40,46)(42,48), (1,7)(3,9)(5,11)(14,20)(16,22)(18,24)(25,37)(26,44)(27,39)(28,46)(29,41)(30,48)(31,43)(32,38)(33,45)(34,40)(35,47)(36,42), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(12,17)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,41,7,47)(2,46,8,40)(3,39,9,45)(4,44,10,38)(5,37,11,43)(6,42,12,48)(13,28,19,34)(14,33,20,27)(15,26,21,32)(16,31,22,25)(17,36,23,30)(18,29,24,35) );
G=PermutationGroup([[(2,19),(4,21),(6,23),(8,13),(10,15),(12,17),(25,43),(26,32),(27,45),(28,34),(29,47),(30,36),(31,37),(33,39),(35,41),(38,44),(40,46),(42,48)], [(1,7),(3,9),(5,11),(14,20),(16,22),(18,24),(25,37),(26,44),(27,39),(28,46),(29,41),(30,48),(31,43),(32,38),(33,45),(34,40),(35,47),(36,42)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,13),(9,14),(10,15),(11,16),(12,17),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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)], [(1,41,7,47),(2,46,8,40),(3,39,9,45),(4,44,10,38),(5,37,11,43),(6,42,12,48),(13,28,19,34),(14,33,20,27),(15,26,21,32),(16,31,22,25),(17,36,23,30),(18,29,24,35)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 12A | 12B | 12C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 12 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | D4⋊2S3 | D4⋊6D6 |
| kernel | C24.46D6 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | C23.11D6 | D4×Dic3 | C23.23D6 | C23.12D6 | D6⋊3D4 | C23.14D6 | C2×C6.D4 | C24⋊4S3 | C3×C22≀C2 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C24.46D6 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,3,10,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
C24.46D6 in GAP, Magma, Sage, TeX
C_2^4._{46}D_6 % in TeX
G:=Group("C2^4.46D6"); // GroupNames label
G:=SmallGroup(192,1152);
// by ID
G=gap.SmallGroup(192,1152);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations