direct product, metabelian, supersoluble, monomial
Aliases: C3×C22.D12, C62.62D4, C62.177C23, D6⋊C4⋊6C6, C6.6(C6×D4), C4⋊Dic3⋊5C6, C2.8(C6×D12), (C2×C6).46D12, C6.94(C2×D12), (C2×C12).232D6, C23.26(S3×C6), C22.4(C3×D12), (C22×Dic3)⋊5C6, (C22×C6).107D6, (C6×C12).191C22, (C2×C62).53C22, C6.115(D4⋊2S3), (C6×Dic3).123C22, C32⋊16(C22.D4), (C2×C4).7(S3×C6), (Dic3×C2×C6)⋊6C2, (C2×C6).5(C3×D4), (C3×D6⋊C4)⋊18C2, C22⋊C4⋊6(C3×S3), (C3×C22⋊C4)⋊4C6, (C2×C12).3(C2×C6), C6.23(C3×C4○D4), (C2×C3⋊D4).5C6, C22.45(S3×C2×C6), (C3×C22⋊C4)⋊14S3, (C3×C4⋊Dic3)⋊29C2, (C3×C6).176(C2×D4), (C6×C3⋊D4).12C2, (S3×C2×C6).56C22, C2.10(C3×D4⋊2S3), (C22×S3).6(C2×C6), (C2×C6).32(C22×C6), (C22×C6).27(C2×C6), C3⋊2(C3×C22.D4), (C3×C6).129(C4○D4), (C32×C22⋊C4)⋊13C2, (C2×C6).310(C22×S3), (C2×Dic3).23(C2×C6), SmallGroup(288,657)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.D12
G = < a,b,c,d,e | a3=b2=c2=d12=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 418 in 173 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3, C6 [×2], C6 [×4], C6 [×10], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C32, Dic3 [×3], C12 [×9], D6 [×3], C2×C6 [×2], C2×C6 [×4], C2×C6 [×12], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×4], C2×C12 [×7], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C22.D4, C3×Dic3 [×3], C3×C12 [×2], S3×C6 [×3], C62, C62 [×2], C62 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4 [×2], C3×C22⋊C4 [×3], C3×C4⋊C4 [×2], C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Dic3, C6×Dic3 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×2], C6×C12 [×2], S3×C2×C6, C2×C62, C22.D12, C3×C22.D4, C3×C4⋊Dic3 [×2], C3×D6⋊C4 [×2], C32×C22⋊C4, Dic3×C2×C6, C6×C3⋊D4, C3×C22.D12
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C4○D4 [×2], C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C22.D4, S3×C6 [×3], C2×D12, D4⋊2S3 [×2], C6×D4, C3×C4○D4 [×2], C3×D12 [×2], S3×C2×C6, C22.D12, C3×C22.D4, C6×D12, C3×D4⋊2S3 [×2], C3×C22.D12
►On 48 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(2 30)(4 32)(6 34)(8 36)(10 26)(12 28)(14 38)(16 40)(18 42)(20 44)(22 46)(24 48)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 21 29 45)(2 44 30 20)(3 19 31 43)(4 42 32 18)(5 17 33 41)(6 40 34 16)(7 15 35 39)(8 38 36 14)(9 13 25 37)(10 48 26 24)(11 23 27 47)(12 46 28 22)
G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (2,30)(4,32)(6,34)(8,36)(10,26)(12,28)(14,38)(16,40)(18,42)(20,44)(22,46)(24,48), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,21,29,45)(2,44,30,20)(3,19,31,43)(4,42,32,18)(5,17,33,41)(6,40,34,16)(7,15,35,39)(8,38,36,14)(9,13,25,37)(10,48,26,24)(11,23,27,47)(12,46,28,22)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (2,30)(4,32)(6,34)(8,36)(10,26)(12,28)(14,38)(16,40)(18,42)(20,44)(22,46)(24,48), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,21,29,45)(2,44,30,20)(3,19,31,43)(4,42,32,18)(5,17,33,41)(6,40,34,16)(7,15,35,39)(8,38,36,14)(9,13,25,37)(10,48,26,24)(11,23,27,47)(12,46,28,22) );
G=PermutationGroup([(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(2,30),(4,32),(6,34),(8,36),(10,26),(12,28),(14,38),(16,40),(18,42),(20,44),(22,46),(24,48)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,25),(10,26),(11,27),(12,28),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(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)], [(1,21,29,45),(2,44,30,20),(3,19,31,43),(4,42,32,18),(5,17,33,41),(6,40,34,16),(7,15,35,39),(8,38,36,14),(9,13,25,37),(10,48,26,24),(11,23,27,47),(12,46,28,22)])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | ··· | 6S | 6T | ··· | 6Y | 6Z | 6AA | 12A | ··· | 12P | 12Q | ··· | 12X | 12Y | 12Z |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | C4○D4 | C3×S3 | D12 | C3×D4 | S3×C6 | S3×C6 | C3×C4○D4 | C3×D12 | D4⋊2S3 | C3×D4⋊2S3 |
| kernel | C3×C22.D12 | C3×C4⋊Dic3 | C3×D6⋊C4 | C32×C22⋊C4 | Dic3×C2×C6 | C6×C3⋊D4 | C22.D12 | C4⋊Dic3 | D6⋊C4 | C3×C22⋊C4 | C22×Dic3 | C2×C3⋊D4 | C3×C22⋊C4 | C62 | C2×C12 | C22×C6 | C3×C6 | C22⋊C4 | C2×C6 | C2×C6 | C2×C4 | C23 | C6 | C22 | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 4 | 2 | 8 | 8 | 2 | 4 |
Matrix representation of C3×C22.D12 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 11 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 11 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 6 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,11,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[8,8,0,0,0,5,0,0,0,0,6,0,0,0,0,11],[5,0,0,0,0,5,0,0,0,0,0,6,0,0,11,0] >;
C3×C22.D12 in GAP, Magma, Sage, TeX
C_3\times C_2^2.D_{12} % in TeX
G:=Group("C3xC2^2.D12"); // GroupNames label
G:=SmallGroup(288,657);
// by ID
G=gap.SmallGroup(288,657);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,555,394,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^12=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations