metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.83D6, (C2×C12)⋊38D4, (C23×C4)⋊10S3, C12⋊7D4⋊51C2, (C23×C12)⋊10C2, D6⋊C4⋊43C22, C12.425(C2×D4), C24⋊4S3⋊15C2, (C2×D12)⋊51C22, C22⋊6(C4○D12), (C2×C6).289C24, C4⋊Dic3⋊65C22, (C22×C4).465D6, C6.135(C22×D4), C12.48D4⋊51C2, (C2×C12).887C23, Dic3⋊C4⋊45C22, C3⋊7(C22.19C24), (C2×Dic6)⋊59C22, (C4×Dic3)⋊59C22, C23.26D6⋊13C2, C23.28D6⋊33C2, C23.245(C22×S3), (C22×C6).418C23, (C23×C6).111C22, C22.304(S3×C23), (C22×S3).127C23, (C22×C12).530C22, (C2×Dic3).151C23, C6.D4.130C22, (C4×C3⋊D4)⋊51C2, (S3×C2×C4)⋊54C22, C6.64(C2×C4○D4), (C2×C4○D12)⋊14C2, (C2×C6)⋊12(C4○D4), (C2×C4)⋊17(C3⋊D4), C2.72(C2×C4○D12), (C2×C6).575(C2×D4), C4.145(C2×C3⋊D4), C2.8(C22×C3⋊D4), C22.35(C2×C3⋊D4), (C2×C4).740(C22×S3), (C2×C3⋊D4).137C22, SmallGroup(192,1350)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 760 in 330 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], S3 [×2], C6, C6 [×2], C6 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], Dic3 [×6], C12 [×4], C12 [×2], D6 [×6], C2×C6, C2×C6 [×6], C2×C6 [×14], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×6], C2×C12 [×10], C22×S3 [×2], C22×C6, C22×C6 [×2], C22×C6 [×6], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×6], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C2×C3⋊D4 [×6], C22×C12 [×2], C22×C12 [×4], C22×C12 [×4], C23×C6, C22.19C24, C12.48D4 [×2], C23.26D6, C4×C3⋊D4 [×4], C23.28D6 [×2], C12⋊7D4 [×2], C24⋊4S3 [×2], C2×C4○D12, C23×C12, C24.83D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C4○D4 [×2], C4○D12 [×4], C2×C3⋊D4 [×6], S3×C23, C22.19C24, C2×C4○D12 [×2], C22×C3⋊D4, C24.83D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, ac=ca, faf-1=ad=da, ae=ea, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
►On 48 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)
(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(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 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(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 46 7 40)(2 39 8 45)(3 44 9 38)(4 37 10 43)(5 42 11 48)(6 47 12 41)(13 26 19 32)(14 31 20 25)(15 36 21 30)(16 29 22 35)(17 34 23 28)(18 27 24 33)
G:=sub<Sym(48)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44), (25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,46,7,40)(2,39,8,45)(3,44,9,38)(4,37,10,43)(5,42,11,48)(6,47,12,41)(13,26,19,32)(14,31,20,25)(15,36,21,30)(16,29,22,35)(17,34,23,28)(18,27,24,33)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44), (25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,37)(36,38), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (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), (1,46,7,40)(2,39,8,45)(3,44,9,38)(4,37,10,43)(5,42,11,48)(6,47,12,41)(13,26,19,32)(14,31,20,25)(15,36,21,30)(16,29,22,35)(17,34,23,28)(18,27,24,33) );
G=PermutationGroup([(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,45),(26,46),(27,47),(28,48),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44)], [(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,37),(36,38)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,37),(36,38)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(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)], [(1,46,7,40),(2,39,8,45),(3,44,9,38),(4,37,10,43),(5,42,11,48),(6,47,12,41),(13,26,19,32),(14,31,20,25),(15,36,21,30),(16,29,22,35),(17,34,23,28),(18,27,24,33)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 11 | 0 | 0 |
| 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,11,0,0,0,0,9,0,0,0,0,3],[0,7,0,0,11,0,0,0,0,0,0,9,0,0,3,0] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D12 |
| kernel | C24.83D6 | C12.48D4 | C23.26D6 | C4×C3⋊D4 | C23.28D6 | C12⋊7D4 | C24⋊4S3 | C2×C4○D12 | C23×C12 | C23×C4 | C2×C12 | C22×C4 | C24 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 6 | 1 | 8 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2^4._{83}D_6 % in TeX
G:=Group("C2^4.83D6"); // GroupNames label
G:=SmallGroup(192,1350);
// by ID
G=gap.SmallGroup(192,1350);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations