metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊11D6, C6.952+ (1+4), C4⋊C4⋊43D6, (C2×C4)⋊5D12, (C2×C12)⋊11D4, D6⋊D4⋊4C2, C4⋊D12⋊3C2, (C4×C12)⋊1C22, D6⋊C4⋊3C22, C4.71(C2×D12), C12⋊D4⋊11C2, C42⋊7S3⋊3C2, C42⋊C2⋊9S3, C2.7(D4○D12), C12.287(C2×D4), (C2×D12)⋊5C22, (C2×C6).69C24, C22⋊C4.93D6, C6.13(C22×D4), (C22×D12)⋊14C2, C22.20(C2×D12), (C22×C4).206D6, C2.15(C22×D12), (C2×C12).144C23, C3⋊1(C22.29C24), (C2×Dic6)⋊51C22, C22.98(S3×C23), (S3×C23).36C22, (C22×S3).19C23, C23.167(C22×S3), (C22×C6).139C23, (C2×Dic3).23C23, (C22×C12).229C22, (S3×C2×C4)⋊1C22, (C2×C6).50(C2×D4), (C2×C4○D12)⋊18C2, (C3×C4⋊C4)⋊53C22, (C3×C42⋊C2)⋊11C2, (C2×C4).149(C22×S3), (C2×C3⋊D4).100C22, (C3×C22⋊C4).101C22, SmallGroup(192,1084)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1128 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], S3 [×6], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×6], D4 [×22], Q8 [×2], C23, C23 [×14], Dic3 [×2], C12 [×4], C12 [×4], D6 [×26], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×19], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×4], D12 [×18], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×6], C22×S3 [×8], C22×C6, C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, D6⋊C4 [×8], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C2×D12 [×12], C2×D12 [×4], C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, S3×C23 [×2], C22.29C24, C4⋊D12 [×2], C42⋊7S3 [×2], D6⋊D4 [×4], C12⋊D4 [×4], C3×C42⋊C2, C22×D12, C2×C4○D12, C42⋊11D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, 2+ (1+4) [×2], C2×D12 [×6], S3×C23, C22.29C24, C22×D12, D4○D12 [×2], C42⋊11D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
►On 48 points
(1 26 4 21)(2 19 5 30)(3 28 6 23)(7 32 46 35)(8 40 47 37)(9 34 48 31)(10 42 43 39)(11 36 44 33)(12 38 45 41)(13 24 16 29)(14 27 17 22)(15 20 18 25)
(1 34 13 41)(2 35 14 42)(3 36 15 37)(4 31 16 38)(5 32 17 39)(6 33 18 40)(7 27 43 19)(8 28 44 20)(9 29 45 21)(10 30 46 22)(11 25 47 23)(12 26 48 24)
(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 40)(2 39)(3 38)(4 37)(5 42)(6 41)(7 27)(8 26)(9 25)(10 30)(11 29)(12 28)(13 33)(14 32)(15 31)(16 36)(17 35)(18 34)(19 43)(20 48)(21 47)(22 46)(23 45)(24 44)
G:=sub<Sym(48)| (1,26,4,21)(2,19,5,30)(3,28,6,23)(7,32,46,35)(8,40,47,37)(9,34,48,31)(10,42,43,39)(11,36,44,33)(12,38,45,41)(13,24,16,29)(14,27,17,22)(15,20,18,25), (1,34,13,41)(2,35,14,42)(3,36,15,37)(4,31,16,38)(5,32,17,39)(6,33,18,40)(7,27,43,19)(8,28,44,20)(9,29,45,21)(10,30,46,22)(11,25,47,23)(12,26,48,24), (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), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,27)(8,26)(9,25)(10,30)(11,29)(12,28)(13,33)(14,32)(15,31)(16,36)(17,35)(18,34)(19,43)(20,48)(21,47)(22,46)(23,45)(24,44)>;
G:=Group( (1,26,4,21)(2,19,5,30)(3,28,6,23)(7,32,46,35)(8,40,47,37)(9,34,48,31)(10,42,43,39)(11,36,44,33)(12,38,45,41)(13,24,16,29)(14,27,17,22)(15,20,18,25), (1,34,13,41)(2,35,14,42)(3,36,15,37)(4,31,16,38)(5,32,17,39)(6,33,18,40)(7,27,43,19)(8,28,44,20)(9,29,45,21)(10,30,46,22)(11,25,47,23)(12,26,48,24), (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), (1,40)(2,39)(3,38)(4,37)(5,42)(6,41)(7,27)(8,26)(9,25)(10,30)(11,29)(12,28)(13,33)(14,32)(15,31)(16,36)(17,35)(18,34)(19,43)(20,48)(21,47)(22,46)(23,45)(24,44) );
G=PermutationGroup([(1,26,4,21),(2,19,5,30),(3,28,6,23),(7,32,46,35),(8,40,47,37),(9,34,48,31),(10,42,43,39),(11,36,44,33),(12,38,45,41),(13,24,16,29),(14,27,17,22),(15,20,18,25)], [(1,34,13,41),(2,35,14,42),(3,36,15,37),(4,31,16,38),(5,32,17,39),(6,33,18,40),(7,27,43,19),(8,28,44,20),(9,29,45,21),(10,30,46,22),(11,25,47,23),(12,26,48,24)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,42),(6,41),(7,27),(8,26),(9,25),(10,30),(11,29),(12,28),(13,33),(14,32),(15,31),(16,36),(17,35),(18,34),(19,43),(20,48),(21,47),(22,46),(23,45),(24,44)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 3 | 6 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 12 |
| 0 | 0 | 12 | 1 | 1 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 3 | 3 | 10 | 10 |
| 0 | 0 | 6 | 10 | 7 | 3 |
G:=sub<GL(6,GF(13))| [3,7,0,0,0,0,6,10,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,11,0,12,0,0,0,0,11,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,12,0,0,1,1,1,1,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,3,6,3,6,0,0,3,10,3,10,0,0,0,0,10,7,0,0,0,0,10,3] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D12 | 2+ (1+4) | D4○D12 |
| kernel | C42⋊11D6 | C4⋊D12 | C42⋊7S3 | D6⋊D4 | C12⋊D4 | C3×C42⋊C2 | C22×D12 | C2×C4○D12 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{11}D_6 % in TeX
G:=Group("C4^2:11D6"); // GroupNames label
G:=SmallGroup(192,1084);
// by ID
G=gap.SmallGroup(192,1084);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations