direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C9, C4⋊C18, C36⋊3C2, C12.3C6, C22⋊2C18, C18.6C22, C3.(C3×D4), C18○(C3×D4), (C3×D4).C3, (C2×C18)⋊1C2, (C2×C6).2C6, C6.6(C2×C6), C2.1(C2×C18), SmallGroup(72,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C9
G = < a,b,c | a9=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C9
►On 36 points
Generators in S36
(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)
(1 34 22 14)(2 35 23 15)(3 36 24 16)(4 28 25 17)(5 29 26 18)(6 30 27 10)(7 31 19 11)(8 32 20 12)(9 33 21 13)
(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)
G:=sub<Sym(36)| (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), (1,34,22,14)(2,35,23,15)(3,36,24,16)(4,28,25,17)(5,29,26,18)(6,30,27,10)(7,31,19,11)(8,32,20,12)(9,33,21,13), (10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)>;
G:=Group( (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), (1,34,22,14)(2,35,23,15)(3,36,24,16)(4,28,25,17)(5,29,26,18)(6,30,27,10)(7,31,19,11)(8,32,20,12)(9,33,21,13), (10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29) );
G=PermutationGroup([[(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)], [(1,34,22,14),(2,35,23,15),(3,36,24,16),(4,28,25,17),(5,29,26,18),(6,30,27,10),(7,31,19,11),(8,32,20,12),(9,33,21,13)], [(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29)]])
D4×C9 is a maximal subgroup of
D4.D9 D4⋊D9 D4⋊2D9 2- 1+4⋊C9
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 12A | 12B | 18A | ··· | 18F | 18G | ··· | 18R | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | D4 | C3×D4 | D4×C9 |
| kernel | D4×C9 | C36 | C2×C18 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 2 | 6 |
Matrix representation of D4×C9 ►in GL2(𝔽19) generated by
| 6 | 0 |
| 0 | 6 |
| 0 | 18 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(19))| [6,0,0,6],[0,1,18,0],[0,1,1,0] >;
D4×C9 in GAP, Magma, Sage, TeX
D_4\times C_9
% in TeX
G:=Group("D4xC9"); // GroupNames label
G:=SmallGroup(72,10);
// by ID
G=gap.SmallGroup(72,10);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-3,141,102]);
// Polycyclic
G:=Group<a,b,c|a^9=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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