direct product, metabelian, supersoluble, monomial
Aliases: C3×C23.26D6, C62.194C23, (C6×C12)⋊14C4, (C2×C12)⋊6C12, C4⋊Dic3⋊17C6, (C2×C12)⋊9Dic3, C12.44(C2×C12), (C4×Dic3)⋊15C6, (C2×C12).447D6, C23.31(S3×C6), C4.15(C6×Dic3), (Dic3×C12)⋊31C2, (C22×C12).42S3, (C22×C12).23C6, C62.110(C2×C4), C6.24(C22×C12), C12.72(C2×Dic3), C6.D4.5C6, (C22×C6).126D6, C6.125(C4○D12), C22.6(C6×Dic3), (C6×C12).326C22, (C2×C62).97C22, C6.44(C22×Dic3), C32⋊17(C42⋊C2), (C6×Dic3).134C22, (C2×C6×C12).14C2, C2.5(Dic3×C2×C6), (C2×C4)⋊4(C3×Dic3), C6.15(C3×C4○D4), C2.4(C3×C4○D12), C22.22(S3×C2×C6), (C2×C4).102(S3×C6), (C2×C6).44(C2×C12), (C3×C4⋊Dic3)⋊35C2, C3⋊4(C3×C42⋊C2), (C2×C12).110(C2×C6), (C3×C12).137(C2×C4), (C2×C6).49(C22×C6), (C22×C6).61(C2×C6), (C22×C4).11(C3×S3), (C2×C6).27(C2×Dic3), (C3×C6).103(C4○D4), (C3×C6).115(C22×C4), (C2×C6).327(C22×S3), (C2×Dic3).34(C2×C6), (C3×C6.D4).10C2, SmallGroup(288,697)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.26D6
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=e5 >
Subgroups: 314 in 179 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×C6, C22×C6, C42⋊C2, C3×Dic3, C3×C12, C62, C62, C62, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C22×C12, C6×Dic3, C6×C12, C6×C12, C2×C62, C23.26D6, C3×C42⋊C2, Dic3×C12, C3×C4⋊Dic3, C3×C6.D4, C2×C6×C12, C3×C23.26D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C4○D4, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C42⋊C2, C3×Dic3, S3×C6, C4○D12, C22×Dic3, C22×C12, C3×C4○D4, C6×Dic3, S3×C2×C6, C23.26D6, C3×C42⋊C2, C3×C4○D12, Dic3×C2×C6, C3×C23.26D6
►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)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)
(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 38 36 16)(2 43 25 21)(3 48 26 14)(4 41 27 19)(5 46 28 24)(6 39 29 17)(7 44 30 22)(8 37 31 15)(9 42 32 20)(10 47 33 13)(11 40 34 18)(12 45 35 23)
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), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40), (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), (1,38,36,16)(2,43,25,21)(3,48,26,14)(4,41,27,19)(5,46,28,24)(6,39,29,17)(7,44,30,22)(8,37,31,15)(9,42,32,20)(10,47,33,13)(11,40,34,18)(12,45,35,23)>;
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), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,37)(22,38)(23,39)(24,40), (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), (1,38,36,16)(2,43,25,21)(3,48,26,14)(4,41,27,19)(5,46,28,24)(6,39,29,17)(7,44,30,22)(8,37,31,15)(9,42,32,20)(10,47,33,13)(11,40,34,18)(12,45,35,23) );
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)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,25),(9,26),(10,27),(11,28),(12,29),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,37),(22,38),(23,39),(24,40)], [(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)], [(1,38,36,16),(2,43,25,21),(3,48,26,14),(4,41,27,19),(5,46,28,24),(6,39,29,17),(7,44,30,22),(8,37,31,15),(9,42,32,20),(10,47,33,13),(11,40,34,18),(12,45,35,23)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 6A | ··· | 6F | 6G | ··· | 6AE | 12A | ··· | 12H | 12I | ··· | 12AJ | 12AK | ··· | 12AZ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
108 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 |
| type | + | + | + | + | + | + | - | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | S3 | Dic3 | D6 | D6 | C4○D4 | C3×S3 | C3×Dic3 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C4○D12 |
| kernel | C3×C23.26D6 | Dic3×C12 | C3×C4⋊Dic3 | C3×C6.D4 | C2×C6×C12 | C23.26D6 | C6×C12 | C4×Dic3 | C4⋊Dic3 | C6.D4 | C22×C12 | C2×C12 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C3×C6 | C22×C4 | C2×C4 | C2×C4 | C23 | C6 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 2 | 1 | 4 | 2 | 8 | 4 | 2 | 8 | 8 | 16 |
Matrix representation of C3×C23.26D6 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 4 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 10 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 2 | 1 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[4,0,0,0,0,10,0,0,0,0,8,0,0,0,0,8],[0,9,0,0,10,0,0,0,0,0,12,2,0,0,12,1] >;
C3×C23.26D6 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{26}D_6 % in TeX
G:=Group("C3xC2^3.26D6"); // GroupNames label
G:=SmallGroup(288,697);
// by ID
G=gap.SmallGroup(288,697);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,1094,9414]);
// 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=e^5>;
// generators/relations