metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊20D6, C6.1242+ 1+4, (C4×S3)⋊4D4, (C2×Q8)⋊21D6, C4.32(S3×D4), C22⋊C4⋊20D6, D6.45(C2×D4), C4.4D4⋊8S3, C12.61(C2×D4), Dic3⋊D4⋊39C2, D6⋊D4⋊23C2, C12⋊3D4⋊24C2, C4⋊D12⋊14C2, (C4×C12)⋊22C22, D6⋊C4⋊23C22, (C2×D4).171D6, (C2×D12)⋊9C22, (C6×Q8)⋊12C22, C6.88(C22×D4), C42⋊2S3⋊19C2, C2.48(D4○D12), (C2×C6).218C24, Dic3.50(C2×D4), C12.23D4⋊21C2, (C2×C12).186C23, Dic3⋊C4⋊55C22, C3⋊4(C22.29C24), (C4×Dic3)⋊35C22, (C6×D4).153C22, (C22×C6).48C23, C23.50(C22×S3), (S3×C23).63C22, C22.239(S3×C23), (C22×S3).213C23, (C2×Dic3).113C23, (C2×S3×D4)⋊16C2, C2.61(C2×S3×D4), (S3×C2×C4)⋊25C22, (C2×Q8⋊3S3)⋊10C2, (C3×C4.4D4)⋊10C2, (C2×C3⋊D4)⋊22C22, (C3×C22⋊C4)⋊28C22, (C2×C4).193(C22×S3), SmallGroup(192,1233)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊20D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=a2b-1, dbd=b-1, dcd=c-1 >
Subgroups: 1104 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×S3, C22×C6, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, S3×D4, Q8⋊3S3, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C22.29C24, C42⋊2S3, C4⋊D12, D6⋊D4, Dic3⋊D4, C12⋊3D4, C12.23D4, C3×C4.4D4, C2×S3×D4, C2×Q8⋊3S3, C42⋊20D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, C22.29C24, C2×S3×D4, D4○D12, C42⋊20D6
►On 48 points
(1 22 9 19)(2 20 7 23)(3 24 8 21)(4 13 11 16)(5 17 12 14)(6 15 10 18)(25 32 38 43)(26 44 39 33)(27 34 40 45)(28 46 41 35)(29 36 42 47)(30 48 37 31)
(1 30 10 40)(2 28 11 38)(3 26 12 42)(4 25 7 41)(5 29 8 39)(6 27 9 37)(13 32 23 35)(14 47 24 44)(15 34 19 31)(16 43 20 46)(17 36 21 33)(18 45 22 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 8)(2 7)(3 9)(4 11)(5 10)(6 12)(14 18)(15 17)(19 21)(22 24)(25 28)(26 27)(29 30)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)(43 46)(44 45)(47 48)
G:=sub<Sym(48)| (1,22,9,19)(2,20,7,23)(3,24,8,21)(4,13,11,16)(5,17,12,14)(6,15,10,18)(25,32,38,43)(26,44,39,33)(27,34,40,45)(28,46,41,35)(29,36,42,47)(30,48,37,31), (1,30,10,40)(2,28,11,38)(3,26,12,42)(4,25,7,41)(5,29,8,39)(6,27,9,37)(13,32,23,35)(14,47,24,44)(15,34,19,31)(16,43,20,46)(17,36,21,33)(18,45,22,48), (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), (1,8)(2,7)(3,9)(4,11)(5,10)(6,12)(14,18)(15,17)(19,21)(22,24)(25,28)(26,27)(29,30)(31,36)(32,35)(33,34)(37,42)(38,41)(39,40)(43,46)(44,45)(47,48)>;
G:=Group( (1,22,9,19)(2,20,7,23)(3,24,8,21)(4,13,11,16)(5,17,12,14)(6,15,10,18)(25,32,38,43)(26,44,39,33)(27,34,40,45)(28,46,41,35)(29,36,42,47)(30,48,37,31), (1,30,10,40)(2,28,11,38)(3,26,12,42)(4,25,7,41)(5,29,8,39)(6,27,9,37)(13,32,23,35)(14,47,24,44)(15,34,19,31)(16,43,20,46)(17,36,21,33)(18,45,22,48), (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), (1,8)(2,7)(3,9)(4,11)(5,10)(6,12)(14,18)(15,17)(19,21)(22,24)(25,28)(26,27)(29,30)(31,36)(32,35)(33,34)(37,42)(38,41)(39,40)(43,46)(44,45)(47,48) );
G=PermutationGroup([[(1,22,9,19),(2,20,7,23),(3,24,8,21),(4,13,11,16),(5,17,12,14),(6,15,10,18),(25,32,38,43),(26,44,39,33),(27,34,40,45),(28,46,41,35),(29,36,42,47),(30,48,37,31)], [(1,30,10,40),(2,28,11,38),(3,26,12,42),(4,25,7,41),(5,29,8,39),(6,27,9,37),(13,32,23,35),(14,47,24,44),(15,34,19,31),(16,43,20,46),(17,36,21,33),(18,45,22,48)], [(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)], [(1,8),(2,7),(3,9),(4,11),(5,10),(6,12),(14,18),(15,17),(19,21),(22,24),(25,28),(26,27),(29,30),(31,36),(32,35),(33,34),(37,42),(38,41),(39,40),(43,46),(44,45),(47,48)]])
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 | 12A | ··· | 12F | 12G | 12H |
| 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 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | D6 | 2+ 1+4 | S3×D4 | D4○D12 |
| kernel | C42⋊20D6 | C42⋊2S3 | C4⋊D12 | D6⋊D4 | Dic3⋊D4 | C12⋊3D4 | C12.23D4 | C3×C4.4D4 | C2×S3×D4 | C2×Q8⋊3S3 | C4.4D4 | C4×S3 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C42⋊20D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,3,6,0,0,0,0,7,10,0,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
C42⋊20D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{20}D_6 % in TeX
G:=Group("C4^2:20D6"); // GroupNames label
G:=SmallGroup(192,1233);
// by ID
G=gap.SmallGroup(192,1233);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,570,297,192,6278]);
// 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=a^2*b^-1,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations