direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×C2×C6, C62⋊6C4, C62.30C22, (C2×C6)⋊5C12, C6⋊2(C2×C12), (C2×C6).49D6, C3⋊2(C22×C12), C23.4(C3×S3), (C22×C6).9C6, C6.9(C22×C6), (C2×C62).4C2, C32⋊7(C22×C4), (C22×C6).12S3, C22.11(S3×C6), C6.48(C22×S3), (C3×C6).27C23, C2.2(S3×C2×C6), (C3×C6)⋊6(C2×C4), (C2×C6).15(C2×C6), SmallGroup(144,166)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C2×C6 |
Generators and relations for Dic3×C2×C6
G = < a,b,c,d | a2=b6=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 184 in 124 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×Dic3, C62, C22×Dic3, C22×C12, C6×Dic3, C2×C62, Dic3×C2×C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×Dic3, S3×C6, C22×Dic3, C22×C12, C6×Dic3, S3×C2×C6, Dic3×C2×C6
►On 48 points
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(13 29)(14 30)(15 25)(16 26)(17 27)(18 28)(37 44)(38 45)(39 46)(40 47)(41 48)(42 43)
(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 13 3 15 5 17)(2 14 4 16 6 18)(7 48 11 46 9 44)(8 43 12 47 10 45)(19 28 21 30 23 26)(20 29 22 25 24 27)(31 38 35 42 33 40)(32 39 36 37 34 41)
(1 44 15 11)(2 45 16 12)(3 46 17 7)(4 47 18 8)(5 48 13 9)(6 43 14 10)(19 42 30 31)(20 37 25 32)(21 38 26 33)(22 39 27 34)(23 40 28 35)(24 41 29 36)
G:=sub<Sym(48)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(13,29)(14,30)(15,25)(16,26)(17,27)(18,28)(37,44)(38,45)(39,46)(40,47)(41,48)(42,43), (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), (1,13,3,15,5,17)(2,14,4,16,6,18)(7,48,11,46,9,44)(8,43,12,47,10,45)(19,28,21,30,23,26)(20,29,22,25,24,27)(31,38,35,42,33,40)(32,39,36,37,34,41), (1,44,15,11)(2,45,16,12)(3,46,17,7)(4,47,18,8)(5,48,13,9)(6,43,14,10)(19,42,30,31)(20,37,25,32)(21,38,26,33)(22,39,27,34)(23,40,28,35)(24,41,29,36)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(13,29)(14,30)(15,25)(16,26)(17,27)(18,28)(37,44)(38,45)(39,46)(40,47)(41,48)(42,43), (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), (1,13,3,15,5,17)(2,14,4,16,6,18)(7,48,11,46,9,44)(8,43,12,47,10,45)(19,28,21,30,23,26)(20,29,22,25,24,27)(31,38,35,42,33,40)(32,39,36,37,34,41), (1,44,15,11)(2,45,16,12)(3,46,17,7)(4,47,18,8)(5,48,13,9)(6,43,14,10)(19,42,30,31)(20,37,25,32)(21,38,26,33)(22,39,27,34)(23,40,28,35)(24,41,29,36) );
G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,19),(7,34),(8,35),(9,36),(10,31),(11,32),(12,33),(13,29),(14,30),(15,25),(16,26),(17,27),(18,28),(37,44),(38,45),(39,46),(40,47),(41,48),(42,43)], [(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)], [(1,13,3,15,5,17),(2,14,4,16,6,18),(7,48,11,46,9,44),(8,43,12,47,10,45),(19,28,21,30,23,26),(20,29,22,25,24,27),(31,38,35,42,33,40),(32,39,36,37,34,41)], [(1,44,15,11),(2,45,16,12),(3,46,17,7),(4,47,18,8),(5,48,13,9),(6,43,14,10),(19,42,30,31),(20,37,25,32),(21,38,26,33),(22,39,27,34),(23,40,28,35),(24,41,29,36)]])
Dic3×C2×C6 is a maximal subgroup of
C62.6Q8 C62.94C23 C62.97C23 C62.56D4 C62⋊3Q8 C62.60D4 C62⋊6D4 S3×C22×C12
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6AI | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 |
| kernel | Dic3×C2×C6 | C6×Dic3 | C2×C62 | C22×Dic3 | C62 | C2×Dic3 | C22×C6 | C2×C6 | C22×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 2 | 8 | 12 | 2 | 16 | 1 | 4 | 3 | 2 | 8 | 6 |
Matrix representation of Dic3×C2×C6 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 9 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 3 |
| 5 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[12,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[5,0,0,0,0,12,0,0,0,0,0,12,0,0,12,0] >;
Dic3×C2×C6 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_2\times C_6 % in TeX
G:=Group("Dic3xC2xC6"); // GroupNames label
G:=SmallGroup(144,166);
// by ID
G=gap.SmallGroup(144,166);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,144,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations