direct product, metacyclic, nilpotent (class 2), monomial
Aliases: D4x3- 1+2, C36:3C6, C62.1C6, C6.15C62, (D4xC9):C3, C9:3(C3xD4), (C2xC18):5C6, C32.(C3xD4), (C3xC12).3C6, C18.7(C2xC6), C12.5(C3xC6), (D4xC32).C3, C3.3(D4xC32), C4:(C2x3- 1+2), (C3xD4).3C32, (C4x3- 1+2):3C2, C22:3(C2x3- 1+2), (C22x3- 1+2):3C2, C2.2(C22x3- 1+2), (C2x3- 1+2).7C22, (C2xC6).8(C3xC6), (C3xC6).14(C2xC6), SmallGroup(216,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4x3- 1+2
G = < a,b,c,d | a4=b2=c9=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Subgroups: 100 in 64 conjugacy classes, 42 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, D4, C9, C32, C12, C12, C2xC6, C2xC6, C18, C18, C3xC6, C3xC6, C3xD4, C3xD4, 3- 1+2, C36, C2xC18, C3xC12, C62, C2x3- 1+2, C2x3- 1+2, D4xC9, D4xC32, C4x3- 1+2, C22x3- 1+2, D4x3- 1+2
Quotients: C1, C2, C3, C22, C6, D4, C32, C2xC6, C3xC6, C3xD4, 3- 1+2, C62, C2x3- 1+2, D4xC32, C22x3- 1+2, D4x3- 1+2
(1 34 27 18)(2 35 19 10)(3 36 20 11)(4 28 21 12)(5 29 22 13)(6 30 23 14)(7 31 24 15)(8 32 25 16)(9 33 26 17)
(10 35)(11 36)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)
G:=sub<Sym(36)| (1,34,27,18)(2,35,19,10)(3,36,20,11)(4,28,21,12)(5,29,22,13)(6,30,23,14)(7,31,24,15)(8,32,25,16)(9,33,26,17), (10,35)(11,36)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34), (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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36)>;
G:=Group( (1,34,27,18)(2,35,19,10)(3,36,20,11)(4,28,21,12)(5,29,22,13)(6,30,23,14)(7,31,24,15)(8,32,25,16)(9,33,26,17), (10,35)(11,36)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34), (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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36) );
G=PermutationGroup([[(1,34,27,18),(2,35,19,10),(3,36,20,11),(4,28,21,12),(5,29,22,13),(6,30,23,14),(7,31,24,15),(8,32,25,16),(9,33,26,17)], [(10,35),(11,36),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34)], [(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)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,25,22),(20,23,26),(29,35,32),(30,33,36)]])
D4x3- 1+2 is a maximal subgroup of
Dic18:C6 D36:C6 Dic18:2C6
55 conjugacy classes
class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 18G | ··· | 18R | 36A | ··· | 36F |
order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
size | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 3 | ··· | 3 | 2 | 2 | 6 | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 |
55 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 6 |
type | + | + | + | + | ||||||||||||
image | C1 | C2 | C2 | C3 | C3 | C6 | C6 | C6 | C6 | D4 | C3xD4 | C3xD4 | 3- 1+2 | C2x3- 1+2 | C2x3- 1+2 | D4x3- 1+2 |
kernel | D4x3- 1+2 | C4x3- 1+2 | C22x3- 1+2 | D4xC9 | D4xC32 | C36 | C2xC18 | C3xC12 | C62 | 3- 1+2 | C9 | C32 | D4 | C4 | C22 | C1 |
# reps | 1 | 1 | 2 | 6 | 2 | 6 | 12 | 2 | 4 | 1 | 6 | 2 | 2 | 2 | 4 | 2 |
Matrix representation of D4x3- 1+2 ►in GL5(F37)
36 | 2 | 0 | 0 | 0 |
36 | 1 | 0 | 0 | 0 |
0 | 0 | 36 | 0 | 0 |
0 | 0 | 0 | 36 | 0 |
0 | 0 | 0 | 0 | 36 |
1 | 0 | 0 | 0 | 0 |
1 | 36 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
10 | 0 | 0 | 0 | 0 |
0 | 10 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 26 |
0 | 0 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 26 | 0 |
0 | 0 | 0 | 0 | 10 |
G:=sub<GL(5,GF(37))| [36,36,0,0,0,2,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,1,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[10,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,26,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,10] >;
D4x3- 1+2 in GAP, Magma, Sage, TeX
D_4\times 3_-^{1+2}
% in TeX
G:=Group("D4xES-(3,1)");
// GroupNames label
G:=SmallGroup(216,78);
// by ID
G=gap.SmallGroup(216,78);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-3,457,338,519]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^9=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations