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

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

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