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), (C4×Dic3)⋊59C22, (C2×Dic6)⋊59C22, C23.28D6⋊33C2, C23.26D6⋊13C2, C22.304(S3×C23), (C23×C6).111C22, C23.245(C22×S3), (C22×C6).418C23, (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
Generators and relations for C24.83D6
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 >
Subgroups: 760 in 330 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C22×C6, C22×C6, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C22×C12, C23×C6, C22.19C24, C12.48D4, C23.26D6, C4×C3⋊D4, C23.28D6, C12⋊7D4, C24⋊4S3, C2×C4○D12, C23×C12, C24.83D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, C4○D12, C2×C3⋊D4, S3×C23, C22.19C24, C2×C4○D12, C22×C3⋊D4, C24.83D6
►On 48 points
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 25)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)
(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 37)(24 38)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 25)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 37)(24 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 23 7 17)(2 16 8 22)(3 21 9 15)(4 14 10 20)(5 19 11 13)(6 24 12 18)(25 44 31 38)(26 37 32 43)(27 42 33 48)(28 47 34 41)(29 40 35 46)(30 45 36 39)
G:=sub<Sym(48)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38), (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), (1,23,7,17)(2,16,8,22)(3,21,9,15)(4,14,10,20)(5,19,11,13)(6,24,12,18)(25,44,31,38)(26,37,32,43)(27,42,33,48)(28,47,34,41)(29,40,35,46)(30,45,36,39)>;
G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,25)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,37)(24,38), (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), (1,23,7,17)(2,16,8,22)(3,21,9,15)(4,14,10,20)(5,19,11,13)(6,24,12,18)(25,44,31,38)(26,37,32,43)(27,42,33,48)(28,47,34,41)(29,40,35,46)(30,45,36,39) );
G=PermutationGroup([[(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,25),(13,45),(14,46),(15,47),(16,48),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44)], [(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,37),(24,38)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,25),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,37),(24,38)], [(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)], [(1,23,7,17),(2,16,8,22),(3,21,9,15),(4,14,10,20),(5,19,11,13),(6,24,12,18),(25,44,31,38),(26,37,32,43),(27,42,33,48),(28,47,34,41),(29,40,35,46),(30,45,36,39)]])
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 |
Matrix representation of C24.83D6 ►in 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] >;
C24.83D6 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