direct product, metabelian, supersoluble, monomial
Aliases: C3×S3×C22⋊C4, C62.171C23, D6⋊C4⋊8C6, D6⋊5(C2×C12), (C2×C12)⋊20D6, C62⋊9(C2×C4), C6.17(C6×D4), C22⋊5(S3×C12), (S3×C6).46D4, D6.10(C3×D4), C6.175(S3×D4), (C22×S3)⋊3C12, (C6×C12)⋊21C22, C6.D4⋊3C6, (S3×C23).2C6, C23.23(S3×C6), C6.6(C22×C12), (C22×C6).104D6, (C6×Dic3)⋊23C22, (C2×C62).47C22, (S3×C2×C4)⋊8C6, (S3×C2×C6)⋊5C4, C2.1(C3×S3×D4), (C2×C4)⋊5(S3×C6), C2.8(S3×C2×C12), (S3×C2×C12)⋊23C2, (C2×C6)⋊3(C2×C12), (C2×C6)⋊16(C4×S3), (C2×C12)⋊6(C2×C6), C3⋊1(C6×C22⋊C4), C6.105(S3×C2×C4), (S3×C6)⋊21(C2×C4), (C3×D6⋊C4)⋊26C2, (C3×C22⋊C4)⋊8C6, (S3×C22×C6).3C2, C32⋊9(C2×C22⋊C4), C22.13(S3×C2×C6), (C2×Dic3)⋊5(C2×C6), (C3×C6).204(C2×D4), (S3×C2×C6).88C22, (C3×C6.D4)⋊5C2, (C2×C6).26(C22×C6), (C3×C6).77(C22×C4), (C22×C6).21(C2×C6), (C32×C22⋊C4)⋊14C2, (C22×S3).16(C2×C6), (C2×C6).304(C22×S3), SmallGroup(288,651)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×S3×C22⋊C4
G = < a,b,c,d,e,f | a3=b3=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, ef=fe >
Subgroups: 706 in 281 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22×C4, C24, C3×S3, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C2×C22⋊C4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, S3×C2×C4, C22×C12, S3×C23, C23×C6, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, S3×C2×C6, S3×C2×C6, C2×C62, S3×C22⋊C4, C6×C22⋊C4, C3×D6⋊C4, C3×C6.D4, C32×C22⋊C4, S3×C2×C12, S3×C22×C6, C3×S3×C22⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C23, C12, D6, C2×C6, C22⋊C4, C22×C4, C2×D4, C3×S3, C4×S3, C2×C12, C3×D4, C22×S3, C22×C6, C2×C22⋊C4, S3×C6, C3×C22⋊C4, S3×C2×C4, S3×D4, C22×C12, C6×D4, S3×C12, S3×C2×C6, S3×C22⋊C4, C6×C22⋊C4, S3×C2×C12, C3×S3×D4, C3×S3×C22⋊C4
►On 48 points
(1 37 23)(2 38 24)(3 39 21)(4 40 22)(5 36 46)(6 33 47)(7 34 48)(8 35 45)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 43 17)(14 44 18)(15 41 19)(16 42 20)
(1 37 23)(2 38 24)(3 39 21)(4 40 22)(5 46 36)(6 47 33)(7 48 34)(8 45 35)(9 29 27)(10 30 28)(11 31 25)(12 32 26)(13 17 43)(14 18 44)(15 19 41)(16 20 42)
(1 47)(2 48)(3 45)(4 46)(5 40)(6 37)(7 38)(8 39)(9 44)(10 41)(11 42)(12 43)(13 26)(14 27)(15 28)(16 25)(17 32)(18 29)(19 30)(20 31)(21 35)(22 36)(23 33)(24 34)
(2 28)(4 26)(5 43)(7 41)(10 38)(12 40)(13 46)(15 48)(17 36)(19 34)(22 32)(24 30)
(1 27)(2 28)(3 25)(4 26)(5 43)(6 44)(7 41)(8 42)(9 37)(10 38)(11 39)(12 40)(13 46)(14 47)(15 48)(16 45)(17 36)(18 33)(19 34)(20 35)(21 31)(22 32)(23 29)(24 30)
(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)
G:=sub<Sym(48)| (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,36,46)(6,33,47)(7,34,48)(8,35,45)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,43,17)(14,44,18)(15,41,19)(16,42,20), (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,46,36)(6,47,33)(7,48,34)(8,45,35)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,17,43)(14,18,44)(15,19,41)(16,20,42), (1,47)(2,48)(3,45)(4,46)(5,40)(6,37)(7,38)(8,39)(9,44)(10,41)(11,42)(12,43)(13,26)(14,27)(15,28)(16,25)(17,32)(18,29)(19,30)(20,31)(21,35)(22,36)(23,33)(24,34), (2,28)(4,26)(5,43)(7,41)(10,38)(12,40)(13,46)(15,48)(17,36)(19,34)(22,32)(24,30), (1,27)(2,28)(3,25)(4,26)(5,43)(6,44)(7,41)(8,42)(9,37)(10,38)(11,39)(12,40)(13,46)(14,47)(15,48)(16,45)(17,36)(18,33)(19,34)(20,35)(21,31)(22,32)(23,29)(24,30), (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)>;
G:=Group( (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,36,46)(6,33,47)(7,34,48)(8,35,45)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,43,17)(14,44,18)(15,41,19)(16,42,20), (1,37,23)(2,38,24)(3,39,21)(4,40,22)(5,46,36)(6,47,33)(7,48,34)(8,45,35)(9,29,27)(10,30,28)(11,31,25)(12,32,26)(13,17,43)(14,18,44)(15,19,41)(16,20,42), (1,47)(2,48)(3,45)(4,46)(5,40)(6,37)(7,38)(8,39)(9,44)(10,41)(11,42)(12,43)(13,26)(14,27)(15,28)(16,25)(17,32)(18,29)(19,30)(20,31)(21,35)(22,36)(23,33)(24,34), (2,28)(4,26)(5,43)(7,41)(10,38)(12,40)(13,46)(15,48)(17,36)(19,34)(22,32)(24,30), (1,27)(2,28)(3,25)(4,26)(5,43)(6,44)(7,41)(8,42)(9,37)(10,38)(11,39)(12,40)(13,46)(14,47)(15,48)(16,45)(17,36)(18,33)(19,34)(20,35)(21,31)(22,32)(23,29)(24,30), (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) );
G=PermutationGroup([[(1,37,23),(2,38,24),(3,39,21),(4,40,22),(5,36,46),(6,33,47),(7,34,48),(8,35,45),(9,29,27),(10,30,28),(11,31,25),(12,32,26),(13,43,17),(14,44,18),(15,41,19),(16,42,20)], [(1,37,23),(2,38,24),(3,39,21),(4,40,22),(5,46,36),(6,47,33),(7,48,34),(8,45,35),(9,29,27),(10,30,28),(11,31,25),(12,32,26),(13,17,43),(14,18,44),(15,19,41),(16,20,42)], [(1,47),(2,48),(3,45),(4,46),(5,40),(6,37),(7,38),(8,39),(9,44),(10,41),(11,42),(12,43),(13,26),(14,27),(15,28),(16,25),(17,32),(18,29),(19,30),(20,31),(21,35),(22,36),(23,33),(24,34)], [(2,28),(4,26),(5,43),(7,41),(10,38),(12,40),(13,46),(15,48),(17,36),(19,34),(22,32),(24,30)], [(1,27),(2,28),(3,25),(4,26),(5,43),(6,44),(7,41),(8,42),(9,37),(10,38),(11,39),(12,40),(13,46),(14,47),(15,48),(16,45),(17,36),(18,33),(19,34),(20,35),(21,31),(22,32),(23,29),(24,30)], [(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)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6S | 6T | ··· | 6AA | 6AB | ··· | 6AG | 6AH | 6AI | 6AJ | 6AK | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | D6 | C3×S3 | C3×D4 | C4×S3 | S3×C6 | S3×C6 | S3×C12 | S3×D4 | C3×S3×D4 |
| kernel | C3×S3×C22⋊C4 | C3×D6⋊C4 | C3×C6.D4 | C32×C22⋊C4 | S3×C2×C12 | S3×C22×C6 | S3×C22⋊C4 | S3×C2×C6 | D6⋊C4 | C6.D4 | C3×C22⋊C4 | S3×C2×C4 | S3×C23 | C22×S3 | C3×C22⋊C4 | S3×C6 | C2×C12 | C22×C6 | C22⋊C4 | D6 | C2×C6 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 8 | 4 | 2 | 2 | 4 | 2 | 16 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 8 | 2 | 4 |
Matrix representation of C3×S3×C22⋊C4 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 11 |
| 0 | 0 | 6 | 0 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,0,8,0,0,0,0,0,6,0,0,11,0] >;
C3×S3×C22⋊C4 in GAP, Magma, Sage, TeX
C_3\times S_3\times C_2^2\rtimes C_4
% in TeX
G:=Group("C3xS3xC2^2:C4"); // GroupNames label
G:=SmallGroup(288,651);
// by ID
G=gap.SmallGroup(288,651);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,555,142,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations