direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.21D6, C62.62D4, C62.177C23, D6⋊C4⋊6C6, C6.6(C6×D4), C4⋊Dic3⋊5C6, C2.8(C6×D12), C6.94(C2×D12), (C2×C6).46D12, (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×C23.21D6
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, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22.D4, C3×Dic3, C3×C12, S3×C6, C62, C62, C62, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C6×Dic3, C6×Dic3, C6×Dic3, C3×C3⋊D4, C6×C12, S3×C2×C6, C2×C62, C23.21D6, C3×C22.D4, C3×C4⋊Dic3, C3×D6⋊C4, C32×C22⋊C4, Dic3×C2×C6, C6×C3⋊D4, C3×C23.21D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C22.D4, S3×C6, C2×D12, D4⋊2S3, C6×D4, C3×C4○D4, C3×D12, S3×C2×C6, C23.21D6, C3×C22.D4, C6×D12, C3×D4⋊2S3, C3×C23.21D6
►On 48 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(2 24)(4 14)(6 16)(8 18)(10 20)(12 22)(26 43)(28 45)(30 47)(32 37)(34 39)(36 41)
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 37)(33 38)(34 39)(35 40)(36 41)
(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 31 23 48)(2 47 24 30)(3 29 13 46)(4 45 14 28)(5 27 15 44)(6 43 16 26)(7 25 17 42)(8 41 18 36)(9 35 19 40)(10 39 20 34)(11 33 21 38)(12 37 22 32)
G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (2,24)(4,14)(6,16)(8,18)(10,20)(12,22)(26,43)(28,45)(30,47)(32,37)(34,39)(36,41), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41), (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), (1,31,23,48)(2,47,24,30)(3,29,13,46)(4,45,14,28)(5,27,15,44)(6,43,16,26)(7,25,17,42)(8,41,18,36)(9,35,19,40)(10,39,20,34)(11,33,21,38)(12,37,22,32)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (2,24)(4,14)(6,16)(8,18)(10,20)(12,22)(26,43)(28,45)(30,47)(32,37)(34,39)(36,41), (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41), (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), (1,31,23,48)(2,47,24,30)(3,29,13,46)(4,45,14,28)(5,27,15,44)(6,43,16,26)(7,25,17,42)(8,41,18,36)(9,35,19,40)(10,39,20,34)(11,33,21,38)(12,37,22,32) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(2,24),(4,14),(6,16),(8,18),(10,20),(12,22),(26,43),(28,45),(30,47),(32,37),(34,39),(36,41)], [(1,23),(2,24),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,37),(33,38),(34,39),(35,40),(36,41)], [(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)], [(1,31,23,48),(2,47,24,30),(3,29,13,46),(4,45,14,28),(5,27,15,44),(6,43,16,26),(7,25,17,42),(8,41,18,36),(9,35,19,40),(10,39,20,34),(11,33,21,38),(12,37,22,32)]])
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×C23.21D6 | C3×C4⋊Dic3 | C3×D6⋊C4 | C32×C22⋊C4 | Dic3×C2×C6 | C6×C3⋊D4 | C23.21D6 | 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×C23.21D6 ►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×C23.21D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{21}D_6 % in TeX
G:=Group("C3xC2^3.21D6"); // 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