metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊9D6, C6.932+ (1+4), C4⋊C4⋊54D6, (C4×D12)⋊5C2, D12⋊26(C2×C4), (C2×D12)⋊18C4, (C4×C12)⋊4C22, C42⋊C2⋊6S3, D6⋊C4⋊60C22, C2.1(D4○D12), (C2×C6).65C24, C6.17(C23×C4), Dic3⋊5D4⋊11C2, D6.4(C22×C4), C4⋊Dic3⋊82C22, C22⋊C4.125D6, (C22×C4).204D6, C12.120(C22×C4), (C2×C12).583C23, C3⋊2(C22.11C24), (C4×Dic3)⋊10C22, (C22×D12).17C2, C22.27(S3×C23), (C2×D12).255C22, C23.26D6⋊24C2, (S3×C23).35C22, (C22×C6).135C23, C23.163(C22×S3), (C22×S3).162C23, (C22×C12).225C22, (C2×Dic3).195C23, C6.D4.94C22, (C2×C4)⋊6(C4×S3), C4.58(S3×C2×C4), (C2×C12)⋊11(C2×C4), (S3×C2×C4)⋊43C22, C2.19(S3×C22×C4), C22.27(S3×C2×C4), (S3×C22⋊C4)⋊25C2, (C3×C4⋊C4)⋊51C22, (C22×S3)⋊7(C2×C4), (C3×C42⋊C2)⋊7C2, (C2×C6).21(C22×C4), (C2×C4).271(C22×S3), (C3×C22⋊C4).135C22, SmallGroup(192,1080)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 936 in 338 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×4], C4 [×8], C22, C22 [×2], C22 [×26], S3 [×8], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×16], C23, C23 [×16], Dic3 [×4], C12 [×4], C12 [×4], D6 [×8], D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], C4×S3 [×8], D12 [×16], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×12], C22×S3 [×4], C22×C6, C2×C22⋊C4 [×4], C42⋊C2, C42⋊C2, C4×D4 [×8], C22×D4, C4×Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4 [×8], C2×D12 [×12], C22×C12, S3×C23 [×2], C22.11C24, C4×D12 [×4], S3×C22⋊C4 [×4], Dic3⋊5D4 [×4], C23.26D6, C3×C42⋊C2, C22×D12, C42⋊9D6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ (1+4) [×2], S3×C2×C4 [×6], S3×C23, C22.11C24, S3×C22×C4, D4○D12 [×2], C42⋊9D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=c-1 >
►On 48 points
(1 38 14 36)(2 39 15 31)(3 40 16 32)(4 41 17 33)(5 42 18 34)(6 37 13 35)(7 27 44 20)(8 28 45 21)(9 29 46 22)(10 30 47 23)(11 25 48 24)(12 26 43 19)
(1 19 4 29)(2 27 5 23)(3 21 6 25)(7 34 47 31)(8 37 48 40)(9 36 43 33)(10 39 44 42)(11 32 45 35)(12 41 46 38)(13 24 16 28)(14 26 17 22)(15 20 18 30)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 36)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)
G:=sub<Sym(48)| (1,38,14,36)(2,39,15,31)(3,40,16,32)(4,41,17,33)(5,42,18,34)(6,37,13,35)(7,27,44,20)(8,28,45,21)(9,29,46,22)(10,30,47,23)(11,25,48,24)(12,26,43,19), (1,19,4,29)(2,27,5,23)(3,21,6,25)(7,34,47,31)(8,37,48,40)(9,36,43,33)(10,39,44,42)(11,32,45,35)(12,41,46,38)(13,24,16,28)(14,26,17,22)(15,20,18,30), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,36)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)>;
G:=Group( (1,38,14,36)(2,39,15,31)(3,40,16,32)(4,41,17,33)(5,42,18,34)(6,37,13,35)(7,27,44,20)(8,28,45,21)(9,29,46,22)(10,30,47,23)(11,25,48,24)(12,26,43,19), (1,19,4,29)(2,27,5,23)(3,21,6,25)(7,34,47,31)(8,37,48,40)(9,36,43,33)(10,39,44,42)(11,32,45,35)(12,41,46,38)(13,24,16,28)(14,26,17,22)(15,20,18,30), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,36)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43) );
G=PermutationGroup([(1,38,14,36),(2,39,15,31),(3,40,16,32),(4,41,17,33),(5,42,18,34),(6,37,13,35),(7,27,44,20),(8,28,45,21),(9,29,46,22),(10,30,47,23),(11,25,48,24),(12,26,43,19)], [(1,19,4,29),(2,27,5,23),(3,21,6,25),(7,34,47,31),(8,37,48,40),(9,36,43,33),(10,39,44,42),(11,32,45,35),(12,41,46,38),(13,24,16,28),(14,26,17,22),(15,20,18,30)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,36),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,38),(14,37),(15,42),(16,41),(17,40),(18,39),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 1 | 1 |
| 0 | 0 | 1 | 0 | 12 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 3 | 10 | 10 | 3 |
| 0 | 0 | 7 | 10 | 6 | 3 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,0,12,0,0,0,0,11,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,12,1,0,0,12,0,12,0,0,0,0,0,1,12,0,0,0,0,1,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,3,7,3,7,0,0,10,10,10,10,0,0,0,0,10,6,0,0,0,0,3,3] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2M | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 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 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | D6 | C4×S3 | 2+ (1+4) | D4○D12 |
| kernel | C42⋊9D6 | C4×D12 | S3×C22⋊C4 | Dic3⋊5D4 | C23.26D6 | C3×C42⋊C2 | C22×D12 | C2×D12 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 16 | 1 | 2 | 2 | 2 | 1 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_9D_6
% in TeX
G:=Group("C4^2:9D6"); // GroupNames label
G:=SmallGroup(192,1080);
// by ID
G=gap.SmallGroup(192,1080);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations