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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

C1C2 — D4×C32
C1C2C6C3×C6C62 — D4×C32
C1C2 — D4×C32
C1C3×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 >

2C2
2C2
2C6
2C6
2C6
2C6
2C6
2C6
2C6
2C6
2C3×C6
2C3×C6

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 2A2B2C3A···3H 4 6A···6H6I···6X12A···12H
order12223···346···66···612···12
size11221···121···12···22···2

45 irreducible representations

dim11111122
type++++
imageC1C2C2C3C6C6D4C3×D4
kernelD4×C32C3×C12C62C3×D4C12C2×C6C32C3
# reps112881618

Matrix representation of D4×C32 in GL3(𝔽13) generated by

300
030
003
,
300
090
009
,
1200
011
01112
,
1200
011
0012
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

Export

Subgroup lattice of D4×C32 in TeX

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