direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.9D6, C62.174C23, D6⋊C4⋊5C6, C4⋊Dic3⋊4C6, D6.4(C3×D4), C6.19(C6×D4), (S3×C6).40D4, C6.178(S3×D4), C23.9(S3×C6), Dic3⋊C4⋊10C6, C6.D4⋊4C6, (C2×C12).231D6, (C22×C6).27D6, C6.117(C4○D12), (C6×C12).243C22, (C2×C62).50C22, C6.113(D4⋊2S3), (C6×Dic3).93C22, C32⋊15(C22.D4), C2.8(C3×S3×D4), (S3×C2×C4)⋊10C6, (S3×C2×C12)⋊25C2, (C3×D6⋊C4)⋊17C2, C22⋊C4⋊3(C3×S3), (C3×C22⋊C4)⋊5C6, (C2×C4).27(S3×C6), C6.22(C3×C4○D4), (C2×C3⋊D4).3C6, C22.42(S3×C2×C6), (C3×C22⋊C4)⋊11S3, (C2×C12).52(C2×C6), (C3×C4⋊Dic3)⋊28C2, C2.10(C3×C4○D12), C2.8(C3×D4⋊2S3), (C3×C6).207(C2×D4), (C6×C3⋊D4).10C2, (S3×C2×C6).90C22, (C3×Dic3⋊C4)⋊29C2, (C3×C6).97(C4○D4), (C32×C22⋊C4)⋊9C2, (C22×C6).24(C2×C6), (C2×C6).29(C22×C6), (C2×Dic3).6(C2×C6), C3⋊1(C3×C22.D4), (C3×C6.D4)⋊21C2, (C22×S3).18(C2×C6), (C2×C6).307(C22×S3), SmallGroup(288,654)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.9D6
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
Subgroups: 418 in 173 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22.D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×C3⋊D4, C22×C12, C6×D4, S3×C12, C6×Dic3, C3×C3⋊D4, C6×C12, S3×C2×C6, C2×C62, C23.9D6, C3×C22.D4, C3×Dic3⋊C4, C3×C4⋊Dic3, C3×D6⋊C4, C3×C6.D4, C32×C22⋊C4, S3×C2×C12, C6×C3⋊D4, C3×C23.9D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3×D4, C22×S3, C22×C6, C22.D4, S3×C6, C4○D12, S3×D4, D4⋊2S3, C6×D4, C3×C4○D4, S3×C2×C6, C23.9D6, C3×C22.D4, C3×C4○D12, C3×S3×D4, C3×D4⋊2S3, C3×C23.9D6
►On 48 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(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 45 41)(38 46 42)(39 47 43)(40 48 44)
(2 15)(4 17)(6 19)(8 21)(10 23)(12 13)(25 39)(26 32)(27 41)(28 34)(29 43)(30 36)(31 45)(33 47)(35 37)(38 44)(40 46)(42 48)
(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 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)
(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 27 7 33)(2 32 8 26)(3 25 9 31)(4 30 10 36)(5 35 11 29)(6 28 12 34)(13 42 19 48)(14 47 20 41)(15 40 21 46)(16 45 22 39)(17 38 23 44)(18 43 24 37)
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (2,15)(4,17)(6,19)(8,21)(10,23)(12,13)(25,39)(26,32)(27,41)(28,34)(29,43)(30,36)(31,45)(33,47)(35,37)(38,44)(40,46)(42,48), (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), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44), (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), (1,27,7,33)(2,32,8,26)(3,25,9,31)(4,30,10,36)(5,35,11,29)(6,28,12,34)(13,42,19,48)(14,47,20,41)(15,40,21,46)(16,45,22,39)(17,38,23,44)(18,43,24,37)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (2,15)(4,17)(6,19)(8,21)(10,23)(12,13)(25,39)(26,32)(27,41)(28,34)(29,43)(30,36)(31,45)(33,47)(35,37)(38,44)(40,46)(42,48), (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), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,13)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44), (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), (1,27,7,33)(2,32,8,26)(3,25,9,31)(4,30,10,36)(5,35,11,29)(6,28,12,34)(13,42,19,48)(14,47,20,41)(15,40,21,46)(16,45,22,39)(17,38,23,44)(18,43,24,37) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(2,15),(4,17),(6,19),(8,21),(10,23),(12,13),(25,39),(26,32),(27,41),(28,34),(29,43),(30,36),(31,45),(33,47),(35,37),(38,44),(40,46),(42,48)], [(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)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,13),(25,45),(26,46),(27,47),(28,48),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44)], [(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)], [(1,27,7,33),(2,32,8,26),(3,25,9,31),(4,30,10,36),(5,35,11,29),(6,28,12,34),(13,42,19,48),(14,47,20,41),(15,40,21,46),(16,45,22,39),(17,38,23,44),(18,43,24,37)]])
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 | ··· | 6O | 6P | ··· | 6W | 6X | 6Y | 6Z | 6AA | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 12S | 12T | 12U | 12V | 12W | 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 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
72 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | C4○D4 | C3×S3 | C3×D4 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C4○D12 | S3×D4 | D4⋊2S3 | C3×S3×D4 | C3×D4⋊2S3 |
| kernel | C3×C23.9D6 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C3×D6⋊C4 | C3×C6.D4 | C32×C22⋊C4 | S3×C2×C12 | C6×C3⋊D4 | C23.9D6 | Dic3⋊C4 | C4⋊Dic3 | D6⋊C4 | C6.D4 | C3×C22⋊C4 | S3×C2×C4 | C2×C3⋊D4 | C3×C22⋊C4 | S3×C6 | C2×C12 | C22×C6 | C3×C6 | C22⋊C4 | D6 | C2×C4 | C23 | C6 | C6 | C2 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 2 | 4 | 8 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C23.9D6 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 6 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 0 | 2 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[6,0,0,0,0,2,0,0,0,0,0,12,0,0,1,0],[0,6,0,0,2,0,0,0,0,0,5,0,0,0,0,5] >;
C3×C23.9D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._9D_6
% in TeX
G:=Group("C3xC2^3.9D6"); // GroupNames label
G:=SmallGroup(288,654);
// by ID
G=gap.SmallGroup(288,654);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,176,590,555,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations