metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊28D6, C6.762+ (1+4), (C4×S3)⋊5D4, (C2×D4)⋊12D6, C4⋊1D4⋊6S3, C4.34(S3×D4), D6.47(C2×D4), C12.65(C2×D4), C23⋊2D6⋊26C2, C12⋊3D4⋊26C2, (C4×C12)⋊26C22, D6⋊C4⋊34C22, (C6×D4)⋊32C22, C6.93(C22×D4), C42⋊2S3⋊23C2, C42⋊7S3⋊25C2, (C2×C6).259C24, Dic3.52(C2×D4), C23.14D6⋊36C2, C2.80(D4⋊6D6), C23.12D6⋊26C2, (C2×C12).635C23, Dic3⋊C4⋊71C22, C3⋊5(C22.29C24), (C4×Dic3)⋊39C22, (C2×Dic6)⋊34C22, (C22×C6).73C23, C23.75(C22×S3), (C2×D12).170C22, C6.D4⋊36C22, (S3×C23).72C22, C22.280(S3×C23), (C22×S3).227C23, (C2×Dic3).134C23, (C22×Dic3)⋊29C22, (C2×S3×D4)⋊19C2, C2.66(C2×S3×D4), (C3×C4⋊1D4)⋊6C2, (C2×D4⋊2S3)⋊20C2, (C2×C3⋊D4)⋊26C22, (S3×C2×C4).138C22, (C2×C4).213(C22×S3), SmallGroup(192,1274)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1040 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×8], C22, C22 [×30], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×22], Q8 [×2], C23 [×4], C23 [×11], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×2], D6 [×16], C2×C6, C2×C6 [×12], C42, C42, C22⋊C4 [×10], C4⋊C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3, C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12, C2×C12 [×2], C3×D4 [×8], C22×S3, C22×S3 [×2], C22×S3 [×8], C22×C6 [×4], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊1D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×6], C6.D4 [×4], C4×C12, C2×Dic6, S3×C2×C4, C2×D12, S3×D4 [×4], D4⋊2S3 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×8], C6×D4 [×2], C6×D4 [×4], S3×C23 [×2], C22.29C24, C42⋊2S3, C42⋊7S3, C23.12D6, C23⋊2D6 [×4], C23.14D6 [×4], C12⋊3D4, C3×C4⋊1D4, C2×S3×D4, C2×D4⋊2S3, C42⋊28D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2+ (1+4) [×2], S3×D4 [×2], S3×C23, C22.29C24, C2×S3×D4, D4⋊6D6 [×2], C42⋊28D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=b-1, dbd=a2b-1, dcd=c-1 >
►On 48 points
(1 48 10 45)(2 46 11 43)(3 44 12 47)(4 23 7 20)(5 21 8 24)(6 19 9 22)(13 34 27 38)(14 39 28 35)(15 36 29 40)(16 41 30 31)(17 32 25 42)(18 37 26 33)
(1 13 8 16)(2 17 9 14)(3 15 7 18)(4 26 12 29)(5 30 10 27)(6 28 11 25)(19 35 43 42)(20 37 44 36)(21 31 45 38)(22 39 46 32)(23 33 47 40)(24 41 48 34)
(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 4)(2 6)(3 5)(7 10)(8 12)(9 11)(13 15)(16 18)(19 43)(20 48)(21 47)(22 46)(23 45)(24 44)(26 30)(27 29)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)
G:=sub<Sym(48)| (1,48,10,45)(2,46,11,43)(3,44,12,47)(4,23,7,20)(5,21,8,24)(6,19,9,22)(13,34,27,38)(14,39,28,35)(15,36,29,40)(16,41,30,31)(17,32,25,42)(18,37,26,33), (1,13,8,16)(2,17,9,14)(3,15,7,18)(4,26,12,29)(5,30,10,27)(6,28,11,25)(19,35,43,42)(20,37,44,36)(21,31,45,38)(22,39,46,32)(23,33,47,40)(24,41,48,34), (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), (1,4)(2,6)(3,5)(7,10)(8,12)(9,11)(13,15)(16,18)(19,43)(20,48)(21,47)(22,46)(23,45)(24,44)(26,30)(27,29)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)>;
G:=Group( (1,48,10,45)(2,46,11,43)(3,44,12,47)(4,23,7,20)(5,21,8,24)(6,19,9,22)(13,34,27,38)(14,39,28,35)(15,36,29,40)(16,41,30,31)(17,32,25,42)(18,37,26,33), (1,13,8,16)(2,17,9,14)(3,15,7,18)(4,26,12,29)(5,30,10,27)(6,28,11,25)(19,35,43,42)(20,37,44,36)(21,31,45,38)(22,39,46,32)(23,33,47,40)(24,41,48,34), (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), (1,4)(2,6)(3,5)(7,10)(8,12)(9,11)(13,15)(16,18)(19,43)(20,48)(21,47)(22,46)(23,45)(24,44)(26,30)(27,29)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38) );
G=PermutationGroup([(1,48,10,45),(2,46,11,43),(3,44,12,47),(4,23,7,20),(5,21,8,24),(6,19,9,22),(13,34,27,38),(14,39,28,35),(15,36,29,40),(16,41,30,31),(17,32,25,42),(18,37,26,33)], [(1,13,8,16),(2,17,9,14),(3,15,7,18),(4,26,12,29),(5,30,10,27),(6,28,11,25),(19,35,43,42),(20,37,44,36),(21,31,45,38),(22,39,46,32),(23,33,47,40),(24,41,48,34)], [(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)], [(1,4),(2,6),(3,5),(7,10),(8,12),(9,11),(13,15),(16,18),(19,43),(20,48),(21,47),(22,46),(23,45),(24,44),(26,30),(27,29),(31,37),(32,42),(33,41),(34,40),(35,39),(36,38)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 6 | 6 |
| 0 | 0 | 11 | 11 | 0 | 6 |
| 0 | 0 | 0 | 9 | 2 | 4 |
| 0 | 0 | 11 | 0 | 2 | 4 |
| 1 | 3 | 0 | 0 | 0 | 0 |
| 8 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 1 | 1 | 11 | 12 |
| 0 | 0 | 5 | 5 | 12 | 0 |
| 0 | 0 | 4 | 5 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 9 | 10 | 12 | 12 |
| 0 | 0 | 8 | 9 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 12 | 12 |
| 0 | 0 | 0 | 12 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,11,0,11,0,0,9,11,9,0,0,0,6,0,2,2,0,0,6,6,4,4],[1,8,0,0,0,0,3,12,0,0,0,0,0,0,0,1,5,4,0,0,0,1,5,5,0,0,12,11,12,12,0,0,1,12,0,0],[12,5,0,0,0,0,0,1,0,0,0,0,0,0,0,12,9,8,0,0,1,1,10,9,0,0,0,0,12,1,0,0,0,0,12,0],[12,5,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,12,12,0,12,0,0,0,0,12,0,0,0,0,0,12,1] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | 2+ (1+4) | S3×D4 | D4⋊6D6 |
| kernel | C42⋊28D6 | C42⋊2S3 | C42⋊7S3 | C23.12D6 | C23⋊2D6 | C23.14D6 | C12⋊3D4 | C3×C4⋊1D4 | C2×S3×D4 | C2×D4⋊2S3 | C4⋊1D4 | C4×S3 | C42 | C2×D4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{28}D_6 % in TeX
G:=Group("C4^2:28D6"); // GroupNames label
G:=SmallGroup(192,1274);
// by ID
G=gap.SmallGroup(192,1274);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,570,297,136,6278]);
// 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,c*b*c^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations