metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊26D6, C6.1382+ 1+4, C4⋊C4⋊34D6, (C4×D12)⋊14C2, (C4×C12)⋊8C22, C42⋊2C2⋊3S3, C42⋊2S3⋊6C2, D6⋊C4⋊63C22, C4.D12⋊40C2, D6⋊Q8⋊41C2, D6⋊D4.3C2, C22⋊C4.77D6, Dic3⋊5D4⋊40C2, D6.26(C4○D4), C23.9D6⋊49C2, D6.D4⋊39C2, C2.63(D4○D12), (C2×C6).249C24, C4⋊Dic3⋊62C22, (C2×C12).603C23, Dic3⋊C4⋊68C22, (C2×Dic6)⋊33C22, (C4×Dic3)⋊58C22, C23.8D6⋊45C2, (C22×C6).63C23, C23.65(C22×S3), C23.11D6⋊45C2, C3⋊9(C22.45C24), (C2×D12).226C22, (S3×C23).69C22, C22.270(S3×C23), (C22×S3).223C23, (C2×Dic3).129C23, C6.D4.65C22, (S3×C2×C4)⋊53C22, C4⋊C4⋊7S3⋊39C2, C2.96(S3×C4○D4), (C3×C4⋊C4)⋊33C22, (S3×C22⋊C4)⋊21C2, C6.207(C2×C4○D4), (C3×C42⋊2C2)⋊4C2, (C2×C4).86(C22×S3), (C2×C3⋊D4).69C22, (C3×C22⋊C4).74C22, SmallGroup(192,1264)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊26D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
Subgroups: 704 in 248 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, S3×C23, C22.45C24, C42⋊2S3, C4×D12, C23.8D6, S3×C22⋊C4, D6⋊D4, C23.9D6, C23.11D6, C4⋊C4⋊7S3, Dic3⋊5D4, D6.D4, D6⋊Q8, C4.D12, C3×C42⋊2C2, C42⋊26D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, S3×C23, C22.45C24, S3×C4○D4, D4○D12, C42⋊26D6
►On 48 points
(1 31 4 16)(2 35 5 14)(3 33 6 18)(7 34 10 13)(8 32 11 17)(9 36 12 15)(19 27 22 43)(20 47 23 25)(21 29 24 45)(26 39 48 42)(28 41 44 38)(30 37 46 40)
(1 22 10 37)(2 20 11 41)(3 24 12 39)(4 19 7 40)(5 23 8 38)(6 21 9 42)(13 46 31 43)(14 25 32 28)(15 48 33 45)(16 27 34 30)(17 44 35 47)(18 29 36 26)
(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 3)(4 6)(7 9)(10 12)(13 18)(14 17)(15 16)(19 39)(20 38)(21 37)(22 42)(23 41)(24 40)(26 30)(27 29)(31 36)(32 35)(33 34)(43 45)(46 48)
G:=sub<Sym(48)| (1,31,4,16)(2,35,5,14)(3,33,6,18)(7,34,10,13)(8,32,11,17)(9,36,12,15)(19,27,22,43)(20,47,23,25)(21,29,24,45)(26,39,48,42)(28,41,44,38)(30,37,46,40), (1,22,10,37)(2,20,11,41)(3,24,12,39)(4,19,7,40)(5,23,8,38)(6,21,9,42)(13,46,31,43)(14,25,32,28)(15,48,33,45)(16,27,34,30)(17,44,35,47)(18,29,36,26), (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), (1,3)(4,6)(7,9)(10,12)(13,18)(14,17)(15,16)(19,39)(20,38)(21,37)(22,42)(23,41)(24,40)(26,30)(27,29)(31,36)(32,35)(33,34)(43,45)(46,48)>;
G:=Group( (1,31,4,16)(2,35,5,14)(3,33,6,18)(7,34,10,13)(8,32,11,17)(9,36,12,15)(19,27,22,43)(20,47,23,25)(21,29,24,45)(26,39,48,42)(28,41,44,38)(30,37,46,40), (1,22,10,37)(2,20,11,41)(3,24,12,39)(4,19,7,40)(5,23,8,38)(6,21,9,42)(13,46,31,43)(14,25,32,28)(15,48,33,45)(16,27,34,30)(17,44,35,47)(18,29,36,26), (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), (1,3)(4,6)(7,9)(10,12)(13,18)(14,17)(15,16)(19,39)(20,38)(21,37)(22,42)(23,41)(24,40)(26,30)(27,29)(31,36)(32,35)(33,34)(43,45)(46,48) );
G=PermutationGroup([[(1,31,4,16),(2,35,5,14),(3,33,6,18),(7,34,10,13),(8,32,11,17),(9,36,12,15),(19,27,22,43),(20,47,23,25),(21,29,24,45),(26,39,48,42),(28,41,44,38),(30,37,46,40)], [(1,22,10,37),(2,20,11,41),(3,24,12,39),(4,19,7,40),(5,23,8,38),(6,21,9,42),(13,46,31,43),(14,25,32,28),(15,48,33,45),(16,27,34,30),(17,44,35,47),(18,29,36,26)], [(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)], [(1,3),(4,6),(7,9),(10,12),(13,18),(14,17),(15,16),(19,39),(20,38),(21,37),(22,42),(23,41),(24,40),(26,30),(27,29),(31,36),(32,35),(33,34),(43,45),(46,48)]])
39 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 | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
39 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 | S3×C4○D4 | D4○D12 |
| kernel | C42⋊26D6 | C42⋊2S3 | C4×D12 | C23.8D6 | S3×C22⋊C4 | D6⋊D4 | C23.9D6 | C23.11D6 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | D6⋊Q8 | C4.D12 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | D6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 8 | 1 | 4 | 2 |
Matrix representation of C42⋊26D6 ►in GL6(𝔽13)
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [0,5,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C42⋊26D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{26}D_6 % in TeX
G:=Group("C4^2:26D6"); // GroupNames label
G:=SmallGroup(192,1264);
// by ID
G=gap.SmallGroup(192,1264);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,570,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations