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## G = D4×C32order 72 = 23·32

### Direct product of C32 and D4

direct product, metacyclic, nilpotent (class 2), monomial

Aliases: D4×C32, C123C6, C621C2, C2.1C62, C4⋊(C3×C6), (C2×C6)⋊3C6, (C3×C12)⋊5C2, C6.8(C2×C6), C222(C3×C6), (C3×C6).16C22, SmallGroup(72,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C32
 Chief series C1 — C2 — C6 — C3×C6 — C62 — D4×C32
 Lower central C1 — C2 — D4×C32
 Upper central C1 — C3×C6 — D4×C32

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   C327D8  C329SD16  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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