metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.42D6, C6.62+ 1+4, C24⋊4S3.C2, D6⋊C4⋊49C22, C23.9D6⋊3C2, (C2×C6).39C24, C4⋊Dic3⋊7C22, C22⋊C4.88D6, (C2×Dic6)⋊4C22, (C22×C4).188D6, C23.8D6⋊2C2, Dic3⋊4D4⋊41C2, C2.10(D4⋊6D6), C12.48D4⋊17C2, (C2×C12).575C23, Dic3⋊C4⋊50C22, Dic3.D4⋊3C2, C23.11D6⋊3C2, (C4×Dic3)⋊49C22, C23.26D6⋊3C2, C23.21D6⋊3C2, (C23×C6).65C22, C22.78(S3×C23), C3⋊1(C22.45C24), C22.17(C4○D12), C23.28D6⋊10C2, C23.16D6⋊25C2, (C22×S3).11C23, C23.232(C22×S3), (C22×C6).129C23, (C2×Dic3).12C23, C22.23(D4⋊2S3), (C22×C12).101C22, C6.D4.140C22, (C22×Dic3).80C22, (C4×C3⋊D4)⋊2C2, (S3×C2×C4)⋊42C22, C6.17(C2×C4○D4), (C2×C22⋊C4)⋊18S3, (C6×C22⋊C4)⋊21C2, C2.19(C2×C4○D12), (C2×C6).40(C4○D4), C2.12(C2×D4⋊2S3), (C2×C3⋊D4).8C22, (C2×C6.D4)⋊18C2, (C2×C4).262(C22×S3), (C3×C22⋊C4).110C22, SmallGroup(192,1054)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.42D6
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, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 600 in 248 conjugacy classes, 99 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, C22×C4, C2×D4, C2×Q8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, 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, C22×C12, C23×C6, C22.45C24, C23.16D6, Dic3.D4, C23.8D6, Dic3⋊4D4, C23.9D6, C23.11D6, C23.21D6, C12.48D4, C23.26D6, C4×C3⋊D4, C23.28D6, C2×C6.D4, C24⋊4S3, C6×C22⋊C4, C24.42D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, D4⋊2S3, S3×C23, C22.45C24, C2×C4○D12, C2×D4⋊2S3, D4⋊6D6, C24.42D6
►On 48 points
(2 17)(4 19)(6 21)(8 23)(10 13)(12 15)(25 31)(26 46)(27 33)(28 48)(29 35)(30 38)(32 40)(34 42)(36 44)(37 43)(39 45)(41 47)
(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(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 26 7 32)(2 31 8 25)(3 36 9 30)(4 29 10 35)(5 34 11 28)(6 27 12 33)(13 37 19 43)(14 42 20 48)(15 47 21 41)(16 40 22 46)(17 45 23 39)(18 38 24 44)
G:=sub<Sym(48)| (2,17)(4,19)(6,21)(8,23)(10,13)(12,15)(25,31)(26,46)(27,33)(28,48)(29,35)(30,38)(32,40)(34,42)(36,44)(37,43)(39,45)(41,47), (25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (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), (1,26,7,32)(2,31,8,25)(3,36,9,30)(4,29,10,35)(5,34,11,28)(6,27,12,33)(13,37,19,43)(14,42,20,48)(15,47,21,41)(16,40,22,46)(17,45,23,39)(18,38,24,44)>;
G:=Group( (2,17)(4,19)(6,21)(8,23)(10,13)(12,15)(25,31)(26,46)(27,33)(28,48)(29,35)(30,38)(32,40)(34,42)(36,44)(37,43)(39,45)(41,47), (25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (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), (1,26,7,32)(2,31,8,25)(3,36,9,30)(4,29,10,35)(5,34,11,28)(6,27,12,33)(13,37,19,43)(14,42,20,48)(15,47,21,41)(16,40,22,46)(17,45,23,39)(18,38,24,44) );
G=PermutationGroup([[(2,17),(4,19),(6,21),(8,23),(10,13),(12,15),(25,31),(26,46),(27,33),(28,48),(29,35),(30,38),(32,40),(34,42),(36,44),(37,43),(39,45),(41,47)], [(25,45),(26,46),(27,47),(28,48),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,37),(36,38)], [(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)], [(1,26,7,32),(2,31,8,25),(3,36,9,30),(4,29,10,35),(5,34,11,28),(6,27,12,33),(13,37,19,43),(14,42,20,48),(15,47,21,41),(16,40,22,46),(17,45,23,39),(18,38,24,44)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | D4⋊2S3 | D4⋊6D6 |
| kernel | C24.42D6 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | C23.11D6 | C23.21D6 | C12.48D4 | C23.26D6 | C4×C3⋊D4 | C23.28D6 | C2×C6.D4 | C24⋊4S3 | C6×C22⋊C4 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C22 | C6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 8 | 8 | 1 | 2 | 2 |
Matrix representation of C24.42D6 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 8 | 11 |
| 0 | 0 | 0 | 5 |
| 0 | 6 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,1,8,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,7,0,0,0,0,8,0,0,0,11,5],[0,2,0,0,6,0,0,0,0,0,8,0,0,0,0,8] >;
C24.42D6 in GAP, Magma, Sage, TeX
C_2^4._{42}D_6 % in TeX
G:=Group("C2^4.42D6"); // GroupNames label
G:=SmallGroup(192,1054);
// by ID
G=gap.SmallGroup(192,1054);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,6278]);
// 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,b*d=d*b,b*e=e*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