metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊8D6, C6.262+ (1+4), C3⋊2(D42), D6⋊6(C2×D4), C3⋊D4⋊4D4, (C2×D4)⋊18D6, C22⋊4(S3×D4), C22≀C2⋊4S3, D6⋊D4⋊9C2, C23⋊2D6⋊4C2, C22⋊C4⋊24D6, Dic3⋊3(C2×D4), (C6×D4)⋊7C22, Dic3⋊D4⋊13C2, C12⋊3D4⋊11C2, D6⋊C4⋊11C22, C6.56(C22×D4), Dic3⋊4D4⋊2C2, C23.14D6⋊2C2, (C2×D12)⋊19C22, (C2×C6).134C24, (C2×C12).28C23, Dic3⋊C4⋊9C22, (S3×C23)⋊7C22, (C23×C6)⋊10C22, C2.28(D4⋊6D6), (C4×Dic3)⋊14C22, C6.D4⋊15C22, (C22×S3).53C23, C23.118(C22×S3), (C22×C6).181C23, C22.155(S3×C23), (C2×Dic3).221C23, (C22×Dic3)⋊13C22, (C2×S3×D4)⋊7C2, (C2×C6)⋊6(C2×D4), C2.29(C2×S3×D4), (S3×C2×C4)⋊7C22, (C3×C22≀C2)⋊5C2, (C22×C3⋊D4)⋊8C2, (C2×C3⋊D4)⋊39C22, (C3×C22⋊C4)⋊5C22, (C2×C4).28(C22×S3), SmallGroup(192,1149)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1344 in 428 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×9], C22, C22 [×4], C22 [×40], S3 [×6], C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×34], C23 [×2], C23 [×2], C23 [×24], Dic3 [×4], Dic3 [×2], C12 [×3], D6 [×4], D6 [×22], C2×C6, C2×C6 [×4], C2×C6 [×14], C42, C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×29], C24, C24 [×3], C4×S3 [×4], D12 [×5], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×8], C3⋊D4 [×16], C2×C12, C2×C12 [×2], C3×D4 [×5], C22×S3 [×4], C22×S3 [×15], C22×C6 [×2], C22×C6 [×2], C22×C6 [×5], C4×D4 [×2], C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×4], C6.D4, C3×C22⋊C4, C3×C22⋊C4 [×2], S3×C2×C4 [×2], C2×D12, C2×D12 [×2], S3×D4 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×10], C2×C3⋊D4 [×8], C6×D4, C6×D4 [×2], S3×C23, S3×C23 [×2], C23×C6, D42, Dic3⋊4D4 [×2], D6⋊D4 [×2], Dic3⋊D4 [×2], C23⋊2D6, C23.14D6 [×2], C12⋊3D4, C3×C22≀C2, C2×S3×D4 [×2], C22×C3⋊D4 [×2], C24⋊8D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C24, C22×S3 [×7], C22×D4 [×2], 2+ (1+4), S3×D4 [×4], S3×C23, D42, C2×S3×D4 [×2], D4⋊6D6, C24⋊8D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
►On 48 points
(1 28)(2 26)(3 30)(4 31)(5 35)(6 33)(7 25)(8 29)(9 27)(10 34)(11 32)(12 36)(13 43)(14 37)(15 45)(16 39)(17 47)(18 41)(19 42)(20 44)(21 38)(22 46)(23 40)(24 48)
(1 13)(2 17)(3 15)(4 16)(5 14)(6 18)(7 19)(8 23)(9 21)(10 22)(11 20)(12 24)(25 42)(26 47)(27 38)(28 43)(29 40)(30 45)(31 39)(32 44)(33 41)(34 46)(35 37)(36 48)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 44)(38 45)(39 46)(40 47)(41 48)(42 43)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 47)(38 48)(39 43)(40 44)(41 45)(42 46)
(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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 47)(38 46)(39 45)(40 44)(41 43)(42 48)
G:=sub<Sym(48)| (1,28)(2,26)(3,30)(4,31)(5,35)(6,33)(7,25)(8,29)(9,27)(10,34)(11,32)(12,36)(13,43)(14,37)(15,45)(16,39)(17,47)(18,41)(19,42)(20,44)(21,38)(22,46)(23,40)(24,48), (1,13)(2,17)(3,15)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24)(25,42)(26,47)(27,38)(28,43)(29,40)(30,45)(31,39)(32,44)(33,41)(34,46)(35,37)(36,48), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,44)(38,45)(39,46)(40,47)(41,48)(42,43), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,47)(38,48)(39,43)(40,44)(41,45)(42,46), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,47)(38,46)(39,45)(40,44)(41,43)(42,48)>;
G:=Group( (1,28)(2,26)(3,30)(4,31)(5,35)(6,33)(7,25)(8,29)(9,27)(10,34)(11,32)(12,36)(13,43)(14,37)(15,45)(16,39)(17,47)(18,41)(19,42)(20,44)(21,38)(22,46)(23,40)(24,48), (1,13)(2,17)(3,15)(4,16)(5,14)(6,18)(7,19)(8,23)(9,21)(10,22)(11,20)(12,24)(25,42)(26,47)(27,38)(28,43)(29,40)(30,45)(31,39)(32,44)(33,41)(34,46)(35,37)(36,48), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,44)(38,45)(39,46)(40,47)(41,48)(42,43), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,34)(26,35)(27,36)(28,31)(29,32)(30,33)(37,47)(38,48)(39,43)(40,44)(41,45)(42,46), (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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,47)(38,46)(39,45)(40,44)(41,43)(42,48) );
G=PermutationGroup([(1,28),(2,26),(3,30),(4,31),(5,35),(6,33),(7,25),(8,29),(9,27),(10,34),(11,32),(12,36),(13,43),(14,37),(15,45),(16,39),(17,47),(18,41),(19,42),(20,44),(21,38),(22,46),(23,40),(24,48)], [(1,13),(2,17),(3,15),(4,16),(5,14),(6,18),(7,19),(8,23),(9,21),(10,22),(11,20),(12,24),(25,42),(26,47),(27,38),(28,43),(29,40),(30,45),(31,39),(32,44),(33,41),(34,46),(35,37),(36,48)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(37,44),(38,45),(39,46),(40,47),(41,48),(42,43)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,34),(26,35),(27,36),(28,31),(29,32),(30,33),(37,47),(38,48),(39,43),(40,44),(41,45),(42,46)], [(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)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(25,33),(26,32),(27,31),(28,36),(29,35),(30,34),(37,47),(38,46),(39,45),(40,44),(41,43),(42,48)])
Matrix representation ►G ⊆ GL6(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | -2 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | -2 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | -2 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | -1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1],[1,0,0,0,0,0,-2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 12A | 12B | 12C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | D6 | 2+ (1+4) | S3×D4 | D4⋊6D6 |
| kernel | C24⋊8D6 | Dic3⋊4D4 | D6⋊D4 | Dic3⋊D4 | C23⋊2D6 | C23.14D6 | C12⋊3D4 | C3×C22≀C2 | C2×S3×D4 | C22×C3⋊D4 | C22≀C2 | C3⋊D4 | C22⋊C4 | C2×D4 | C24 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 8 | 3 | 3 | 1 | 1 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_2^4\rtimes_8D_6
% in TeX
G:=Group("C2^4:8D6"); // GroupNames label
G:=SmallGroup(192,1149);
// by ID
G=gap.SmallGroup(192,1149);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,f*b*f=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations