direct product, metabelian, soluble, monomial, A-group
Aliases: C6×C3.A4, C62.6C6, (C22×C6)⋊C9, (C2×C6)⋊2C18, C3.2(C6×A4), C6.5(C3×A4), (C3×C6).5A4, C23⋊2(C3×C9), C22⋊2(C3×C18), C32.3(C2×A4), (C2×C62).1C3, (C22×C6).3C32, (C2×C6).3(C3×C6), SmallGroup(216,105)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C6×C3.A4 |
Generators and relations for C6×C3.A4
G = < a,b,c,d,e | a6=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Subgroups: 136 in 64 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C6, C23, C9, C32, C2×C6, C2×C6, C2×C6, C18, C3×C6, C3×C6, C22×C6, C22×C6, C3×C9, C3.A4, C62, C62, C3×C18, C2×C3.A4, C2×C62, C3×C3.A4, C6×C3.A4
Quotients: C1, C2, C3, C6, C9, C32, A4, C18, C3×C6, C2×A4, C3×C9, C3.A4, C3×A4, C3×C18, C2×C3.A4, C6×A4, C3×C3.A4, C6×C3.A4
►On 54 points
Generators in S54
(1 33 24 52 15 41)(2 34 25 53 16 42)(3 35 26 54 17 43)(4 36 27 46 18 44)(5 28 19 47 10 45)(6 29 20 48 11 37)(7 30 21 49 12 38)(8 31 22 50 13 39)(9 32 23 51 14 40)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)
(2 53)(3 54)(5 47)(6 48)(8 50)(9 51)(10 28)(11 29)(13 31)(14 32)(16 34)(17 35)(19 45)(20 37)(22 39)(23 40)(25 42)(26 43)
(1 52)(3 54)(4 46)(6 48)(7 49)(9 51)(11 29)(12 30)(14 32)(15 33)(17 35)(18 36)(20 37)(21 38)(23 40)(24 41)(26 43)(27 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 49 50 51 52 53 54)
G:=sub<Sym(54)| (1,33,24,52,15,41)(2,34,25,53,16,42)(3,35,26,54,17,43)(4,36,27,46,18,44)(5,28,19,47,10,45)(6,29,20,48,11,37)(7,30,21,49,12,38)(8,31,22,50,13,39)(9,32,23,51,14,40), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,53)(3,54)(5,47)(6,48)(8,50)(9,51)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35)(19,45)(20,37)(22,39)(23,40)(25,42)(26,43), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(11,29)(12,30)(14,32)(15,33)(17,35)(18,36)(20,37)(21,38)(23,40)(24,41)(26,43)(27,44), (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)(46,47,48,49,50,51,52,53,54)>;
G:=Group( (1,33,24,52,15,41)(2,34,25,53,16,42)(3,35,26,54,17,43)(4,36,27,46,18,44)(5,28,19,47,10,45)(6,29,20,48,11,37)(7,30,21,49,12,38)(8,31,22,50,13,39)(9,32,23,51,14,40), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,53)(3,54)(5,47)(6,48)(8,50)(9,51)(10,28)(11,29)(13,31)(14,32)(16,34)(17,35)(19,45)(20,37)(22,39)(23,40)(25,42)(26,43), (1,52)(3,54)(4,46)(6,48)(7,49)(9,51)(11,29)(12,30)(14,32)(15,33)(17,35)(18,36)(20,37)(21,38)(23,40)(24,41)(26,43)(27,44), (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)(46,47,48,49,50,51,52,53,54) );
G=PermutationGroup([[(1,33,24,52,15,41),(2,34,25,53,16,42),(3,35,26,54,17,43),(4,36,27,46,18,44),(5,28,19,47,10,45),(6,29,20,48,11,37),(7,30,21,49,12,38),(8,31,22,50,13,39),(9,32,23,51,14,40)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(2,53),(3,54),(5,47),(6,48),(8,50),(9,51),(10,28),(11,29),(13,31),(14,32),(16,34),(17,35),(19,45),(20,37),(22,39),(23,40),(25,42),(26,43)], [(1,52),(3,54),(4,46),(6,48),(7,49),(9,51),(11,29),(12,30),(14,32),(15,33),(17,35),(18,36),(20,37),(21,38),(23,40),(24,41),(26,43),(27,44)], [(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),(46,47,48,49,50,51,52,53,54)]])
C6×C3.A4 is a maximal subgroup of
C62.10Dic3
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6H | 6I | ··· | 6X | 9A | ··· | 9R | 18A | ··· | 18R |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 | A4 | C2×A4 | C3.A4 | C3×A4 | C2×C3.A4 | C6×A4 |
| kernel | C6×C3.A4 | C3×C3.A4 | C2×C3.A4 | C2×C62 | C3.A4 | C62 | C22×C6 | C2×C6 | C3×C6 | C32 | C6 | C6 | C3 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 | 1 | 1 | 6 | 2 | 6 | 2 |
Matrix representation of C6×C3.A4 ►in GL5(𝔽19)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(19))| [12,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
C6×C3.A4 in GAP, Magma, Sage, TeX
C_6\times C_3.A_4
% in TeX
G:=Group("C6xC3.A4"); // GroupNames label
G:=SmallGroup(216,105);
// by ID
G=gap.SmallGroup(216,105);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,115,1630,2927]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations