direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.28D6, C62.120D4, C62.197C23, D6⋊C4⋊2C6, C6.42(C6×D4), Dic3⋊C4⋊3C6, (C22×C12)⋊6C6, (C22×C12)⋊7S3, C6.D4⋊6C6, (C2×C12).353D6, C23.33(S3×C6), (C22×C6).128D6, C6.127(C4○D12), (C6×C12).283C22, (C2×C62).100C22, (C6×Dic3).98C22, C32⋊21(C22.D4), (C2×C6×C12)⋊2C2, C2.6(C6×C3⋊D4), (C3×D6⋊C4)⋊33C2, (C2×C4).68(S3×C6), (C22×C4)⋊7(C3×S3), C6.17(C3×C4○D4), (C2×C6).46(C3×D4), (C2×C3⋊D4).6C6, C22.55(S3×C2×C6), (C2×C12).92(C2×C6), C2.18(C3×C4○D12), (C3×C6).253(C2×D4), (C6×C3⋊D4).13C2, C6.143(C2×C3⋊D4), (S3×C2×C6).59C22, C22.9(C3×C3⋊D4), (C3×Dic3⋊C4)⋊35C2, (C2×C6).62(C3⋊D4), (C22×S3).9(C2×C6), (C22×C6).64(C2×C6), (C2×C6).52(C22×C6), C3⋊4(C3×C22.D4), (C3×C6).105(C4○D4), (C3×C6.D4)⋊23C2, (C2×C6).330(C22×S3), (C2×Dic3).10(C2×C6), SmallGroup(288,700)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.28D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >
Subgroups: 410 in 183 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, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C3×C6, 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, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×Dic3, C6×Dic3, C3×C3⋊D4, C6×C12, C6×C12, S3×C2×C6, C2×C62, C23.28D6, C3×C22.D4, C3×Dic3⋊C4, C3×D6⋊C4, C3×C6.D4, C6×C3⋊D4, C2×C6×C12, C3×C23.28D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C22.D4, S3×C6, C4○D12, C2×C3⋊D4, C6×D4, C3×C4○D4, C3×C3⋊D4, S3×C2×C6, C23.28D6, C3×C22.D4, C3×C4○D12, C6×C3⋊D4, C3×C23.28D6
►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)
(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 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)
(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 42 13 26)(2 25 14 41)(3 40 15 36)(4 35 16 39)(5 38 17 34)(6 33 18 37)(7 48 19 32)(8 31 20 47)(9 46 21 30)(10 29 22 45)(11 44 23 28)(12 27 24 43)
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), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46), (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), (1,42,13,26)(2,25,14,41)(3,40,15,36)(4,35,16,39)(5,38,17,34)(6,33,18,37)(7,48,19,32)(8,31,20,47)(9,46,21,30)(10,29,22,45)(11,44,23,28)(12,27,24,43)>;
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), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46), (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), (1,42,13,26)(2,25,14,41)(3,40,15,36)(4,35,16,39)(5,38,17,34)(6,33,18,37)(7,48,19,32)(8,31,20,47)(9,46,21,30)(10,29,22,45)(11,44,23,28)(12,27,24,43) );
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)], [(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,47),(26,48),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46)], [(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)], [(1,42,13,26),(2,25,14,41),(3,40,15,36),(4,35,16,39),(5,38,17,34),(6,33,18,37),(7,48,19,32),(8,31,20,47),(9,46,21,30),(10,29,22,45),(11,44,23,28),(12,27,24,43)]])
90 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 | ··· | 6AE | 6AF | 6AG | 12A | ··· | 12AF | 12AG | ··· | 12AL |
| 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 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 2 | ··· | 2 | 12 | ··· | 12 |
90 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 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | C4○D4 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C3⋊D4 | C3×C4○D12 |
| kernel | C3×C23.28D6 | C3×Dic3⋊C4 | C3×D6⋊C4 | C3×C6.D4 | C6×C3⋊D4 | C2×C6×C12 | C23.28D6 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C22×C12 | C22×C12 | C62 | C2×C12 | C22×C6 | C3×C6 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C23 | C6 | C6 | C22 | 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 | 8 | 16 |
Matrix representation of C3×C23.28D6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 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 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 8 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 0 | 11 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,0,8,0,0,0,0,2,0,0,0,0,7],[0,8,0,0,8,0,0,0,0,0,0,11,0,0,6,0] >;
C3×C23.28D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{28}D_6 % in TeX
G:=Group("C3xC2^3.28D6"); // GroupNames label
G:=SmallGroup(288,700);
// by ID
G=gap.SmallGroup(288,700);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,590,268,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^5>;
// generators/relations