metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊19D6, C6.212+ (1+4), C4⋊C4⋊50D6, (C4×D4)⋊22S3, D6⋊D4⋊7C2, (D4×C12)⋊24C2, C22⋊C4⋊49D6, (C22×C4)⋊17D6, Dic3⋊D4⋊10C2, C23⋊2D6⋊21C2, C12⋊7D4⋊11C2, (C4×C12)⋊28C22, D6⋊C4⋊31C22, (C2×D4).221D6, D6.D4⋊8C2, C42⋊3S3⋊10C2, C42⋊7S3⋊28C2, C2.17(D4○D12), (C2×C6).104C24, C4⋊Dic3⋊10C22, (C2×Dic6)⋊7C22, C23.14D6⋊27C2, C2.22(D4⋊6D6), (C2×C12).162C23, Dic3⋊C4⋊33C22, (C22×C12)⋊11C22, C23.11D6⋊9C2, Dic3.D4⋊9C2, C3⋊2(C22.32C24), (C4×Dic3)⋊53C22, (C6×D4).308C22, (C2×D12).27C22, C22.6(C4○D12), C23.28D6⋊2C2, C6.D4⋊10C22, (S3×C23).42C22, (C22×S3).38C23, C23.111(C22×S3), C22.129(S3×C23), (C22×C6).174C23, (C2×Dic3).45C23, (C22×Dic3).99C22, C4⋊C4⋊S3⋊8C2, (C2×D6⋊C4)⋊35C2, (C4×C3⋊D4)⋊46C2, (S3×C2×C4)⋊49C22, C6.46(C2×C4○D4), (C3×C4⋊C4)⋊62C22, C2.53(C2×C4○D12), (C2×C3⋊D4)⋊5C22, (C2×C6).17(C4○D4), (C3×C22⋊C4)⋊58C22, (C2×C4).162(C22×S3), SmallGroup(192,1119)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 744 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×2], C22 [×18], S3 [×3], C6 [×3], C6 [×3], C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], Dic3 [×5], C12 [×5], D6 [×13], C2×C6, C2×C6 [×2], C2×C6 [×5], C42, C42, C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic6, C4×S3, D12 [×2], C2×Dic3 [×5], C2×Dic3, C3⋊D4 [×5], C2×C12 [×5], C2×C12 [×2], C3×D4 [×2], C22×S3 [×3], C22×S3 [×4], C22×C6 [×2], C2×C22⋊C4, C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3, D6⋊C4 [×10], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12 [×2], C22×Dic3, C2×C3⋊D4 [×4], C22×C12 [×2], C6×D4, S3×C23, C22.32C24, C42⋊7S3, C42⋊3S3, Dic3.D4, D6⋊D4, Dic3⋊D4, C23.11D6, D6.D4, C4⋊C4⋊S3, C2×D6⋊C4, C4×C3⋊D4, C23.28D6, C12⋊7D4, C23⋊2D6, C23.14D6, D4×C12, C42⋊19D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4) [×2], C4○D12 [×2], S3×C23, C22.32C24, C2×C4○D12, D4⋊6D6, D4○D12, C42⋊19D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
►On 48 points
(1 32 19 35)(2 36 20 33)(3 34 21 31)(4 39 17 42)(5 37 18 40)(6 41 16 38)(7 25 10 28)(8 29 11 26)(9 27 12 30)(13 44 23 47)(14 48 24 45)(15 46 22 43)
(1 16 24 9)(2 17 22 7)(3 18 23 8)(4 15 10 20)(5 13 11 21)(6 14 12 19)(25 36 42 43)(26 31 37 44)(27 32 38 45)(28 33 39 46)(29 34 40 47)(30 35 41 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 20)(2 19)(3 21)(4 6)(7 9)(10 12)(13 23)(14 22)(15 24)(16 17)(25 41)(26 40)(27 39)(28 38)(29 37)(30 42)(31 44)(32 43)(33 48)(34 47)(35 46)(36 45)
G:=sub<Sym(48)| (1,32,19,35)(2,36,20,33)(3,34,21,31)(4,39,17,42)(5,37,18,40)(6,41,16,38)(7,25,10,28)(8,29,11,26)(9,27,12,30)(13,44,23,47)(14,48,24,45)(15,46,22,43), (1,16,24,9)(2,17,22,7)(3,18,23,8)(4,15,10,20)(5,13,11,21)(6,14,12,19)(25,36,42,43)(26,31,37,44)(27,32,38,45)(28,33,39,46)(29,34,40,47)(30,35,41,48), (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), (1,20)(2,19)(3,21)(4,6)(7,9)(10,12)(13,23)(14,22)(15,24)(16,17)(25,41)(26,40)(27,39)(28,38)(29,37)(30,42)(31,44)(32,43)(33,48)(34,47)(35,46)(36,45)>;
G:=Group( (1,32,19,35)(2,36,20,33)(3,34,21,31)(4,39,17,42)(5,37,18,40)(6,41,16,38)(7,25,10,28)(8,29,11,26)(9,27,12,30)(13,44,23,47)(14,48,24,45)(15,46,22,43), (1,16,24,9)(2,17,22,7)(3,18,23,8)(4,15,10,20)(5,13,11,21)(6,14,12,19)(25,36,42,43)(26,31,37,44)(27,32,38,45)(28,33,39,46)(29,34,40,47)(30,35,41,48), (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), (1,20)(2,19)(3,21)(4,6)(7,9)(10,12)(13,23)(14,22)(15,24)(16,17)(25,41)(26,40)(27,39)(28,38)(29,37)(30,42)(31,44)(32,43)(33,48)(34,47)(35,46)(36,45) );
G=PermutationGroup([(1,32,19,35),(2,36,20,33),(3,34,21,31),(4,39,17,42),(5,37,18,40),(6,41,16,38),(7,25,10,28),(8,29,11,26),(9,27,12,30),(13,44,23,47),(14,48,24,45),(15,46,22,43)], [(1,16,24,9),(2,17,22,7),(3,18,23,8),(4,15,10,20),(5,13,11,21),(6,14,12,19),(25,36,42,43),(26,31,37,44),(27,32,38,45),(28,33,39,46),(29,34,40,47),(30,35,41,48)], [(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)], [(1,20),(2,19),(3,21),(4,6),(7,9),(10,12),(13,23),(14,22),(15,24),(16,17),(25,41),(26,40),(27,39),(28,38),(29,37),(30,42),(31,44),(32,43),(33,48),(34,47),(35,46),(36,45)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 11 | 9 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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))| [11,4,0,0,0,0,9,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 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 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | D4⋊6D6 | D4○D12 |
| kernel | C42⋊19D6 | C42⋊7S3 | C42⋊3S3 | Dic3.D4 | D6⋊D4 | Dic3⋊D4 | C23.11D6 | D6.D4 | C4⋊C4⋊S3 | C2×D6⋊C4 | C4×C3⋊D4 | C23.28D6 | C12⋊7D4 | C23⋊2D6 | C23.14D6 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C22 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{19}D_6 % in TeX
G:=Group("C4^2:19D6"); // GroupNames label
G:=SmallGroup(192,1119);
// by ID
G=gap.SmallGroup(192,1119);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations