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G = C32⋊2D8  order 144 = 24·32

1st semidirect product of C32 and D8 acting via D8/C4=C22

metabelian, supersoluble, monomial

Aliases: C32⋊2D8, D12⋊2S3, C12.9D6, C4.8S32, (C3×C6).6D4, C3⋊2(D4⋊S3), (C3×D12)⋊1C2, C32⋊4C8⋊1C2, C6.7(C3⋊D4), (C3×C12).1C22, C2.3(D6⋊S3), SmallGroup(144,56)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C32⋊2D8
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C32⋊2D8
Lower central
C32 — C3×C6 — C3×C12 — C32⋊2D8
Upper central
C1 — C2 — C4

Generators and relations for C32⋊2D8
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

C1
C2
12C2
12C2
C3
C3
2C3
C4
6C22
6C22
C6
C6
2C6
4S3
4S3
12C6
12C6
C32
3D4
3D4
9C8
C12
C12
2D6
2C12
2D6
6C2×C6
6C2×C6
C3×C6
4C3×S3
4C3×S3
9D8
D12
D12
3C3×D4
3C3⋊C8
3C3⋊C8
3C3×D4
6C3⋊C8
C3×C12
2S3×C6
2S3×C6
3D4⋊S3
3D4⋊S3
C3×D12
C32⋊4C8
C3×D12
C32⋊2D8

Character table of C32⋊2D8

 class 12A2B2C3A3B3C46A6B6C6D6E6F6G8A8B12A12B12C12D
 size 11121222422241212121218184444
ρ1111111111111111111111    trivial
ρ211-1-11111111-1-1-1-1111111    linear of order 2
ρ3111-11111111-1-111-1-11111    linear of order 2
ρ411-11111111111-1-1-1-11111    linear of order 2
ρ52200222-2222000000-2-2-2-2    orthogonal lifted from D4
ρ622-202-1-12-12-10011002-1-1-1    orthogonal lifted from D6
ρ722202-1-12-12-100-1-1002-1-1-1    orthogonal lifted from S3
ρ82202-12-122-1-1-1-10000-12-1-1    orthogonal lifted from S3
ρ9220-2-12-122-1-1110000-12-1-1    orthogonal lifted from D6
ρ102-2002220-2-2-20000√2-√20000    orthogonal lifted from D8
ρ112-2002220-2-2-20000-√2√20000    orthogonal lifted from D8
ρ1222002-1-1-2-12-100-√-3√-300-2111    complex lifted from C3⋊D4
ρ1322002-1-1-2-12-100√-3-√-300-2111    complex lifted from C3⋊D4
ρ142200-12-1-22-1-1√-3-√-300001-211    complex lifted from C3⋊D4
ρ152200-12-1-22-1-1-√-3√-300001-211    complex lifted from C3⋊D4
ρ164-4004-2-202-420000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ174-400-24-20-4220000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ184400-2-214-2-21000000-2-211    orthogonal lifted from S32
ρ194400-2-21-4-2-2100000022-1-1    symplectic lifted from D6⋊S3, Schur index 2
ρ204-400-2-21022-1000000003i-3i    complex faithful
ρ214-400-2-21022-100000000-3i3i    complex faithful

Smallest permutation representation of C32⋊2D8
►On 48 points
Generators in S48
(1 35 46)(2 47 36)(3 37 48)(4 41 38)(5 39 42)(6 43 40)(7 33 44)(8 45 34)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)

(1 46 35)(2 36 47)(3 48 37)(4 38 41)(5 42 39)(6 40 43)(7 44 33)(8 34 45)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)

(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)

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

C32⋊2D8 is a maximal subgroup of
 C32⋊D16  C32⋊SD32  C24⋊4D6  C24⋊6D6  D12.2D6  D12.30D6  D12⋊20D6  S3×D4⋊S3  D12⋊9D6  D12⋊6D6  D12.12D6  D36⋊S3  He3⋊3D8  C33⋊6D8  C33⋊9D8
C32⋊2D8 is a maximal quotient of
 C32⋊2D16  D24.S3  C32⋊2Q32  D12⋊3Dic3  C12.8Dic6  D36⋊S3  He3⋊2D8  C33⋊6D8  C33⋊9D8

Matrix representation of C32⋊2D8 ►in GL4(𝔽5) generated by

4011
1421
2444
2111
,
4340
0401
1203
0400
,
0023
1114
0433
0011
,
2112
4231
1122
1044
G:=sub<GL(4,GF(5))| [4,1,2,2,0,4,4,1,1,2,4,1,1,1,4,1],[4,0,1,0,3,4,2,4,4,0,0,0,0,1,3,0],[0,1,0,0,0,1,4,0,2,1,3,1,3,4,3,1],[2,4,1,1,1,2,1,0,1,3,2,4,2,1,2,4] >;
 

C32⋊2D8 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_2D_8
 
% in TeX
 
G:=Group("C3^2:2D8");
 
// GroupNames label
 
G:=SmallGroup(144,56);
 
// by ID
 
G=gap.SmallGroup(144,56);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,218,116,50,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊2D8 in TeX
Character table of C32⋊2D8 in TeX

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