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G = C32⋊D16  order 288 = 25·32

The semidirect product of C32 and D16 acting via D16/C4=D4

non-abelian, soluble, monomial

Aliases: C32⋊D16, C4.1S3≀C2, (C3×C6).1D8, (C3×C12).5D4, C32⋊2D8⋊1C2, C2.3(C32⋊D8), C32⋊2C16⋊1C2, C32⋊4C8.1C22, SmallGroup(288,382)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C32⋊4C8 — C32⋊D16
Chief series
C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2D8 — C32⋊D16
Lower central
C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊D16
Upper central
C1 — C2 — C4

Generators and relations for C32⋊D16
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

C1
C2
24C2
24C2
2C3
2C3
C4
12C22
12C22
2C6
2C6
8S3
8S3
24C6
24C6
C32
6D4
6D4
9C8
2C12
2C12
4D6
4D6
12C2×C6
12C2×C6
C3×C6
8C3×S3
8C3×S3
9C16
9D8
9D8
2D12
2D12
6C3⋊C8
6C3×D4
6C3⋊C8
6C3×D4
C3×C12
4S3×C6
4S3×C6
9D16
6D4⋊S3
6D4⋊S3
C32⋊4C8
2C3×D12
2C3×D12
C32⋊2D8
C32⋊2D8
C32⋊2C16
C32⋊D16

Character table of C32⋊D16

 class 12A2B2C3A3B46A6B6C6D6E6F8A8B12A12B16A16B16C16D
 size 112424442442424242418188818181818
ρ1111111111111111111111    trivial
ρ211-1-111111-1-1-1-111111111    linear of order 2
ρ311-1111111-111-11111-1-1-1-1    linear of order 2
ρ4111-1111111-1-111111-1-1-1-1    linear of order 2
ρ52200222220000-2-2220000    orthogonal lifted from D4
ρ6220022-222000000-2-2√2-√2√2-√2    orthogonal lifted from D8
ρ7220022-222000000-2-2-√2√2-√2√2    orthogonal lifted from D8
ρ82-200220-2-20000-√2√200-ζ165+ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169    orthogonal lifted from D16
ρ92-200220-2-20000-√2√200ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169    orthogonal lifted from D16
ρ102-200220-2-20000√2-√200-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163    orthogonal lifted from D16
ρ112-200220-2-20000√2-√200ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163    orthogonal lifted from D16
ρ12440-21-241-2011000-210000    orthogonal lifted from S3≀C2
ρ134420-214-21-100-1001-20000    orthogonal lifted from S3≀C2
ρ1444-20-214-211001001-20000    orthogonal lifted from S3≀C2
ρ1544021-241-20-1-1000-210000    orthogonal lifted from S3≀C2
ρ1644001-2-41-20√-3-√-30002-10000    complex lifted from C32⋊D8
ρ174400-21-4-21-√-300√-300-120000    complex lifted from C32⋊D8
ρ1844001-2-41-20-√-3√-30002-10000    complex lifted from C32⋊D8
ρ194400-21-4-21√-300-√-300-120000    complex lifted from C32⋊D8
ρ208-8002-40-24000000000000    orthogonal faithful, Schur index 2
ρ218-800-4204-2000000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of C32⋊D16
►On 48 points
Generators in S48
(1 21 48)(3 34 23)(5 25 36)(7 38 27)(9 29 40)(11 42 31)(13 17 44)(15 46 19)

(2 33 22)(4 24 35)(6 37 26)(8 28 39)(10 41 30)(12 32 43)(14 45 18)(16 20 47)

(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 37 38 39 40 41 42 43 44 45 46 47 48)

(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 36)(18 35)(19 34)(20 33)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)
 
G:=sub<Sym(48)| (1,21,48)(3,34,23)(5,25,36)(7,38,27)(9,29,40)(11,42,31)(13,17,44)(15,46,19), (2,33,22)(4,24,35)(6,37,26)(8,28,39)(10,41,30)(12,32,43)(14,45,18)(16,20,47), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,36)(18,35)(19,34)(20,33)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)>;
 
G:=Group( (1,21,48)(3,34,23)(5,25,36)(7,38,27)(9,29,40)(11,42,31)(13,17,44)(15,46,19), (2,33,22)(4,24,35)(6,37,26)(8,28,39)(10,41,30)(12,32,43)(14,45,18)(16,20,47), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,36)(18,35)(19,34)(20,33)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37) );
 
G=PermutationGroup([[(1,21,48),(3,34,23),(5,25,36),(7,38,27),(9,29,40),(11,42,31),(13,17,44),(15,46,19)], [(2,33,22),(4,24,35),(6,37,26),(8,28,39),(10,41,30),(12,32,43),(14,45,18),(16,20,47)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,36),(18,35),(19,34),(20,33),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37)]])
 

Matrix representation of C32⋊D16 ►in GL6(𝔽97)

100000
010000
0096100
0096000
0073010
0073001
,
100000
010000
001000
000100
000001
0024249696
,
95710000
26950000
0000196
00737321
003434240
003435240
,
100000
0960000
000100
001000
001616960
001616096

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,73,73,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,24,0,0,0,1,0,24,0,0,0,0,0,96,0,0,0,0,1,96],[95,26,0,0,0,0,71,95,0,0,0,0,0,0,0,73,34,34,0,0,0,73,34,35,0,0,1,2,24,24,0,0,96,1,0,0],[1,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,16,16,0,0,1,0,16,16,0,0,0,0,96,0,0,0,0,0,0,96] >;
 

C32⋊D16 in GAP, Magma, Sage, TeX 

C_3^2\rtimes D_{16}
 
% in TeX
 
G:=Group("C3^2:D16");
 
// GroupNames label
 
G:=SmallGroup(288,382);
 
// by ID
 
G=gap.SmallGroup(288,382);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,254,135,142,675,346,80,2693,2028,691,797,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊D16 in TeX
Character table of C32⋊D16 in TeX

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