direct product, metacyclic, nilpotent (class 2), monomial
Aliases: D4×C32, C12⋊3C6, C62⋊1C2, C2.1C62, C4⋊(C3×C6), (C2×C6)⋊3C6, (C3×C12)⋊5C2, C6.8(C2×C6), C22⋊2(C3×C6), (C3×C6).16C22, SmallGroup(72,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C32
G = < a,b,c,d | a3=b3=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D4×C32
►On 36 points
Generators in S36
(1 20 29)(2 17 30)(3 18 31)(4 19 32)(5 16 25)(6 13 26)(7 14 27)(8 15 28)(9 36 21)(10 33 22)(11 34 23)(12 35 24)
(1 14 10)(2 15 11)(3 16 12)(4 13 9)(5 24 31)(6 21 32)(7 22 29)(8 23 30)(17 28 34)(18 25 35)(19 26 36)(20 27 33)
(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 4)(6 8)(9 11)(13 15)(17 19)(21 23)(26 28)(30 32)(34 36)
G:=sub<Sym(36)| (1,20,29)(2,17,30)(3,18,31)(4,19,32)(5,16,25)(6,13,26)(7,14,27)(8,15,28)(9,36,21)(10,33,22)(11,34,23)(12,35,24), (1,14,10)(2,15,11)(3,16,12)(4,13,9)(5,24,31)(6,21,32)(7,22,29)(8,23,30)(17,28,34)(18,25,35)(19,26,36)(20,27,33), (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), (2,4)(6,8)(9,11)(13,15)(17,19)(21,23)(26,28)(30,32)(34,36)>;
G:=Group( (1,20,29)(2,17,30)(3,18,31)(4,19,32)(5,16,25)(6,13,26)(7,14,27)(8,15,28)(9,36,21)(10,33,22)(11,34,23)(12,35,24), (1,14,10)(2,15,11)(3,16,12)(4,13,9)(5,24,31)(6,21,32)(7,22,29)(8,23,30)(17,28,34)(18,25,35)(19,26,36)(20,27,33), (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), (2,4)(6,8)(9,11)(13,15)(17,19)(21,23)(26,28)(30,32)(34,36) );
G=PermutationGroup([[(1,20,29),(2,17,30),(3,18,31),(4,19,32),(5,16,25),(6,13,26),(7,14,27),(8,15,28),(9,36,21),(10,33,22),(11,34,23),(12,35,24)], [(1,14,10),(2,15,11),(3,16,12),(4,13,9),(5,24,31),(6,21,32),(7,22,29),(8,23,30),(17,28,34),(18,25,35),(19,26,36),(20,27,33)], [(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)], [(2,4),(6,8),(9,11),(13,15),(17,19),(21,23),(26,28),(30,32),(34,36)]])
D4×C32 is a maximal subgroup of
C32⋊7D8 C32⋊9SD16 C12.D6
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4 | 6A | ··· | 6H | 6I | ··· | 6X | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | C3×D4 |
| kernel | D4×C32 | C3×C12 | C62 | C3×D4 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 2 | 8 | 8 | 16 | 1 | 8 |
Matrix representation of D4×C32 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 3 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 12 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 11 | 12 |
| 12 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 12 |
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[3,0,0,0,9,0,0,0,9],[12,0,0,0,1,11,0,1,12],[12,0,0,0,1,0,0,1,12] >;
D4×C32 in GAP, Magma, Sage, TeX
D_4\times C_3^2
% in TeX
G:=Group("D4xC3^2"); // GroupNames label
G:=SmallGroup(72,37);
// by ID
G=gap.SmallGroup(72,37);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-2,381]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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