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, (C4×Dic3)⋊58C22, (C2×Dic6)⋊33C22, C23.8D6⋊45C2, C23.65(C22×S3), (C22×C6).63C23, 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), (S3×C22⋊C4)⋊21C2, (C3×C4⋊C4)⋊33C22, 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
Subgroups: 704 in 248 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×11], C22, C22 [×18], S3 [×5], C6 [×3], C6, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], Dic3 [×5], C12 [×6], D6 [×4], D6 [×11], C2×C6, C2×C6 [×3], C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×5], C22×C4 [×5], C2×D4 [×3], C2×Q8, C24, Dic6, C4×S3 [×7], D12 [×4], C2×Dic3 [×5], C3⋊D4, C2×C12 [×6], C22×S3 [×3], C22×S3 [×5], C22×C6, C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2, C42⋊2C2, C4×Dic3 [×2], Dic3⋊C4 [×3], C4⋊Dic3 [×2], D6⋊C4 [×9], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6, S3×C2×C4 [×5], C2×D12 [×2], C2×C3⋊D4, S3×C23, C22.45C24, C42⋊2S3, C4×D12, C23.8D6, S3×C22⋊C4 [×2], D6⋊D4, C23.9D6, C23.11D6, C4⋊C4⋊7S3, Dic3⋊5D4, D6.D4 [×2], D6⋊Q8, C4.D12, C3×C42⋊2C2, C42⋊26D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), S3×C23, C22.45C24, S3×C4○D4 [×2], D4○D12, C42⋊26D6
Generators and relations
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 >
►On 48 points
(1 38 4 28)(2 42 5 26)(3 40 6 30)(7 41 10 25)(8 39 11 29)(9 37 12 27)(13 46 16 34)(14 32 17 44)(15 48 18 36)(19 31 22 43)(20 47 23 35)(21 33 24 45)
(1 22 10 13)(2 20 11 17)(3 24 12 15)(4 19 7 16)(5 23 8 14)(6 21 9 18)(25 46 38 43)(26 35 39 32)(27 48 40 45)(28 31 41 34)(29 44 42 47)(30 33 37 36)
(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 21)(14 20)(15 19)(16 24)(17 23)(18 22)(25 30)(26 29)(27 28)(31 33)(34 36)(37 38)(39 42)(40 41)(43 45)(46 48)
G:=sub<Sym(48)| (1,38,4,28)(2,42,5,26)(3,40,6,30)(7,41,10,25)(8,39,11,29)(9,37,12,27)(13,46,16,34)(14,32,17,44)(15,48,18,36)(19,31,22,43)(20,47,23,35)(21,33,24,45), (1,22,10,13)(2,20,11,17)(3,24,12,15)(4,19,7,16)(5,23,8,14)(6,21,9,18)(25,46,38,43)(26,35,39,32)(27,48,40,45)(28,31,41,34)(29,44,42,47)(30,33,37,36), (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)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,30)(26,29)(27,28)(31,33)(34,36)(37,38)(39,42)(40,41)(43,45)(46,48)>;
G:=Group( (1,38,4,28)(2,42,5,26)(3,40,6,30)(7,41,10,25)(8,39,11,29)(9,37,12,27)(13,46,16,34)(14,32,17,44)(15,48,18,36)(19,31,22,43)(20,47,23,35)(21,33,24,45), (1,22,10,13)(2,20,11,17)(3,24,12,15)(4,19,7,16)(5,23,8,14)(6,21,9,18)(25,46,38,43)(26,35,39,32)(27,48,40,45)(28,31,41,34)(29,44,42,47)(30,33,37,36), (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)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22)(25,30)(26,29)(27,28)(31,33)(34,36)(37,38)(39,42)(40,41)(43,45)(46,48) );
G=PermutationGroup([(1,38,4,28),(2,42,5,26),(3,40,6,30),(7,41,10,25),(8,39,11,29),(9,37,12,27),(13,46,16,34),(14,32,17,44),(15,48,18,36),(19,31,22,43),(20,47,23,35),(21,33,24,45)], [(1,22,10,13),(2,20,11,17),(3,24,12,15),(4,19,7,16),(5,23,8,14),(6,21,9,18),(25,46,38,43),(26,35,39,32),(27,48,40,45),(28,31,41,34),(29,44,42,47),(30,33,37,36)], [(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),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22),(25,30),(26,29),(27,28),(31,33),(34,36),(37,38),(39,42),(40,41),(43,45),(46,48)])
Matrix representation ►G ⊆ 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] >;
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 |
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