metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊7D6, C6.252+ (1+4), (C2×D4)⋊5D6, C22⋊C4⋊5D6, D6.2(C2×D4), C22≀C2⋊3S3, C23⋊2D6⋊3C2, (C6×D4)⋊6C22, (C22×S3)⋊6D4, Dic3⋊D4⋊12C2, D6⋊3D4⋊12C2, C24⋊4S3⋊6C2, D6⋊C4⋊10C22, C3⋊2(C23⋊3D4), (C23×C6)⋊9C22, C6.55(C22×D4), C22.40(S3×D4), C23.9D6⋊12C2, (C2×D12)⋊18C22, (C2×C6).133C24, (C2×C12).27C23, Dic3⋊C4⋊8C22, (S3×C23)⋊6C22, C4⋊Dic3⋊25C22, (C22×C6).8C23, C2.27(D4⋊6D6), C23.23D6⋊4C2, C22.D12⋊9C2, C6.D4⋊14C22, C22.154(S3×C23), C23.117(C22×S3), (C2×Dic3).60C23, (C22×S3).182C23, (C22×Dic3)⋊12C22, (C2×S3×D4)⋊6C2, C2.28(C2×S3×D4), (S3×C2×C4)⋊6C22, (S3×C22⋊C4)⋊2C2, (C2×C6).53(C2×D4), (C3×C22≀C2)⋊4C2, (C22×C3⋊D4)⋊7C2, (C2×C3⋊D4)⋊38C22, (C3×C22⋊C4)⋊4C22, (C2×C4).27(C22×S3), SmallGroup(192,1148)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1072 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C22, C22 [×2], C22 [×34], S3 [×5], C6, C6 [×2], C6 [×5], C2×C4, C2×C4 [×2], C2×C4 [×11], D4 [×20], C23 [×2], C23 [×2], C23 [×17], Dic3 [×5], C12 [×3], D6 [×4], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×15], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×17], C24, C24 [×2], C4×S3 [×4], D12 [×2], C2×Dic3, C2×Dic3 [×4], C2×Dic3 [×2], C3⋊D4 [×14], C2×C12, C2×C12 [×2], C3×D4 [×4], C22×S3 [×3], C22×S3 [×4], C22×S3 [×6], C22×C6 [×2], C22×C6 [×2], C22×C6 [×4], C2×C22⋊C4, C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4, C6.D4 [×4], C3×C22⋊C4, C3×C22⋊C4 [×2], S3×C2×C4, S3×C2×C4 [×2], C2×D12, S3×D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×C3⋊D4 [×6], C2×C3⋊D4 [×4], C6×D4, C6×D4 [×2], S3×C23 [×2], C23×C6, C23⋊3D4, S3×C22⋊C4, C23.9D6 [×2], Dic3⋊D4 [×2], C22.D12, C23.23D6, C23⋊2D6 [×2], D6⋊3D4 [×2], C24⋊4S3, C3×C22≀C2, C2×S3×D4, C22×C3⋊D4, C24⋊7D6
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, C23⋊3D4, C2×S3×D4, D4⋊6D6 [×2], C24⋊7D6
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, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
►On 48 points
(1 37)(2 24)(3 39)(4 20)(5 41)(6 22)(7 27)(8 44)(9 29)(10 46)(11 25)(12 48)(13 45)(14 30)(15 47)(16 26)(17 43)(18 28)(19 33)(21 35)(23 31)(32 38)(34 40)(36 42)
(1 17)(2 44)(3 13)(4 46)(5 15)(6 48)(7 31)(8 24)(9 33)(10 20)(11 35)(12 22)(14 40)(16 42)(18 38)(19 29)(21 25)(23 27)(26 36)(28 32)(30 34)(37 43)(39 45)(41 47)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 33)(20 34)(21 35)(22 36)(23 31)(24 32)
(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 16)(8 15)(9 14)(10 13)(11 18)(12 17)(19 20)(21 24)(22 23)(25 44)(26 43)(27 48)(28 47)(29 46)(30 45)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)
G:=sub<Sym(48)| (1,37)(2,24)(3,39)(4,20)(5,41)(6,22)(7,27)(8,44)(9,29)(10,46)(11,25)(12,48)(13,45)(14,30)(15,47)(16,26)(17,43)(18,28)(19,33)(21,35)(23,31)(32,38)(34,40)(36,42), (1,17)(2,44)(3,13)(4,46)(5,15)(6,48)(7,31)(8,24)(9,33)(10,20)(11,35)(12,22)(14,40)(16,42)(18,38)(19,29)(21,25)(23,27)(26,36)(28,32)(30,34)(37,43)(39,45)(41,47), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(19,33)(20,34)(21,35)(22,36)(23,31)(24,32), (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)(7,16)(8,15)(9,14)(10,13)(11,18)(12,17)(19,20)(21,24)(22,23)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(31,36)(32,35)(33,34)(37,42)(38,41)(39,40)>;
G:=Group( (1,37)(2,24)(3,39)(4,20)(5,41)(6,22)(7,27)(8,44)(9,29)(10,46)(11,25)(12,48)(13,45)(14,30)(15,47)(16,26)(17,43)(18,28)(19,33)(21,35)(23,31)(32,38)(34,40)(36,42), (1,17)(2,44)(3,13)(4,46)(5,15)(6,48)(7,31)(8,24)(9,33)(10,20)(11,35)(12,22)(14,40)(16,42)(18,38)(19,29)(21,25)(23,27)(26,36)(28,32)(30,34)(37,43)(39,45)(41,47), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,39)(20,40)(21,41)(22,42)(23,37)(24,38)(25,47)(26,48)(27,43)(28,44)(29,45)(30,46), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(19,33)(20,34)(21,35)(22,36)(23,31)(24,32), (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)(7,16)(8,15)(9,14)(10,13)(11,18)(12,17)(19,20)(21,24)(22,23)(25,44)(26,43)(27,48)(28,47)(29,46)(30,45)(31,36)(32,35)(33,34)(37,42)(38,41)(39,40) );
G=PermutationGroup([(1,37),(2,24),(3,39),(4,20),(5,41),(6,22),(7,27),(8,44),(9,29),(10,46),(11,25),(12,48),(13,45),(14,30),(15,47),(16,26),(17,43),(18,28),(19,33),(21,35),(23,31),(32,38),(34,40),(36,42)], [(1,17),(2,44),(3,13),(4,46),(5,15),(6,48),(7,31),(8,24),(9,33),(10,20),(11,35),(12,22),(14,40),(16,42),(18,38),(19,29),(21,25),(23,27),(26,36),(28,32),(30,34),(37,43),(39,45),(41,47)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16),(19,39),(20,40),(21,41),(22,42),(23,37),(24,38),(25,47),(26,48),(27,43),(28,44),(29,45),(30,46)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,45),(14,46),(15,47),(16,48),(17,43),(18,44),(19,33),(20,34),(21,35),(22,36),(23,31),(24,32)], [(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),(7,16),(8,15),(9,14),(10,13),(11,18),(12,17),(19,20),(21,24),(22,23),(25,44),(26,43),(27,48),(28,47),(29,46),(30,45),(31,36),(32,35),(33,34),(37,42),(38,41),(39,40)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 6 | 0 | 0 | 0 | 0 |
| 9 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,9,0,0,0,0,6,5,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,2,9,0,0,0,0,4,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,12,1,0,0,0,0,12,0,0,0],[12,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,12,0,0,0,0,0,12,1,0,0] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | 4D | ··· | 4H | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 12A | 12B | 12C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | D6 | 2+ (1+4) | S3×D4 | D4⋊6D6 |
| kernel | C24⋊7D6 | S3×C22⋊C4 | C23.9D6 | Dic3⋊D4 | C22.D12 | C23.23D6 | C23⋊2D6 | D6⋊3D4 | C24⋊4S3 | C3×C22≀C2 | C2×S3×D4 | C22×C3⋊D4 | C22≀C2 | C22×S3 | C22⋊C4 | C2×D4 | C24 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2^4\rtimes_7D_6
% in TeX
G:=Group("C2^4:7D6"); // GroupNames label
G:=SmallGroup(192,1148);
// by ID
G=gap.SmallGroup(192,1148);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,297,6278]);
// 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,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,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