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G = C32⋊D8⋊C2  order 288 = 25·32

3rd semidirect product of C32⋊D8 and C2 acting faithfully

non-abelian, soluble, monomial

Aliases: C32⋊D8⋊3C2, C4.14S3≀C2, D6⋊D6⋊6C2, (C3×C12).11D4, D6.D6⋊2C2, C32⋊1(C8⋊C22), D6⋊S3⋊8C22, C32⋊2C8⋊1C22, C32⋊2Q8⋊7C22, C32⋊2SD16⋊1C2, C3⋊Dic3.2C23, C32⋊M4(2)⋊4C2, C2.8(C2×S3≀C2), (C3×C6).5(C2×D4), (C2×C3⋊S3).29D4, (C4×C3⋊S3).30C22, SmallGroup(288,872)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊Dic3 — C32⋊D8⋊C2
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — D6⋊S3 — C32⋊D8 — C32⋊D8⋊C2
Lower central
C32 — C3×C6 — C3⋊Dic3 — C32⋊D8⋊C2
Upper central
C1 — C2 — C4

Generators and relations for C32⋊D8⋊C2
 G = < a,b,c,d,e | a3=b3=c8=d2=e2=1, ab=ba, cac-1=b, dad=eae=cbc-1=a-1, bd=db, ebe=b-1, dcd=c-1, ece=c5, ede=c4d >

Subgroups: 688 in 115 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C4○D12, S3×D4, C32⋊2C8, D6⋊S3, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C3×D12, C4×C3⋊S3, C2×S32, C32⋊D8, C32⋊2SD16, C32⋊M4(2), D6.D6, D6⋊D6, C32⋊D8⋊C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C8⋊C22, S3≀C2, C2×S3≀C2, C32⋊D8⋊C2

Character table of C32⋊D8⋊C2

 class 12A2B2C2D2E3A3B4A4B4C6A6B6C6D6E6F8A8B12A12B12C12D12E
 size 11121212184421218441212242436364481212
ρ1111111111111111111111111    trivial
ρ211-111-111-1-111111-111-1-1-1-1-1-1    linear of order 2
ρ311-11-1-111-11111-1-1-11-11-1-1-111    linear of order 2
ρ41111-11111-1111-1-111-1-1111-1-1    linear of order 2
ρ5111-1-1-111-11111-1-11-11-1-1-1-111    linear of order 2
ρ611-1-1-11111-1111-1-1-1-111111-1-1    linear of order 2
ρ711-1-111111111111-1-1-1-111111    linear of order 2
ρ8111-11-111-1-1111111-1-11-1-1-1-1-1    linear of order 2
ρ922000222-20-222000000-2-2-200    orthogonal lifted from D4
ρ1022000-22220-22200000022200    orthogonal lifted from D4
ρ1144-2-200-214001-2001100-2-2100    orthogonal lifted from S3≀C2
ρ12442-200-21-4001-200-110022-100    orthogonal lifted from C2×S3≀C2
ρ134-4000044000-4-400000000000    orthogonal lifted from C8⋊C22
ρ14442200-214001-200-1-100-2-2100    orthogonal lifted from S3≀C2
ρ154400-201-2-420-21110000-1-12-1-1    orthogonal lifted from C2×S3≀C2
ρ164400201-2420-21-1-1000011-2-1-1    orthogonal lifted from S3≀C2
ρ1744-2200-21-4001-2001-10022-100    orthogonal lifted from C2×S3≀C2
ρ184400-201-24-20-2111000011-211    orthogonal lifted from S3≀C2
ρ194400201-2-4-20-21-1-10000-1-1211    orthogonal lifted from C2×S3≀C2
ρ204-400001-20002-1-√-3√-30000-3i3i0-√3√3    complex faithful
ρ214-400001-20002-1-√-3√-300003i-3i0√3-√3    complex faithful
ρ224-400001-20002-1√-3-√-30000-3i3i0√3-√3    complex faithful
ρ234-400001-20002-1√-3-√-300003i-3i0-√3√3    complex faithful
ρ248-80000-42000-2400000000000    orthogonal faithful

Permutation representations of C32⋊D8⋊C2
►On 24 points - transitive group 24T658
Generators in S24
(1 15 21)(2 16 22)(3 23 9)(4 24 10)(5 11 17)(6 12 18)(7 19 13)(8 20 14)

(1 15 21)(2 22 16)(3 23 9)(4 10 24)(5 11 17)(6 18 12)(7 19 13)(8 14 20)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)

(1 5)(3 7)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)
 
G:=sub<Sym(24)| (1,15,21)(2,16,22)(3,23,9)(4,24,10)(5,11,17)(6,12,18)(7,19,13)(8,20,14), (1,15,21)(2,22,16)(3,23,9)(4,10,24)(5,11,17)(6,18,12)(7,19,13)(8,14,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21), (1,5)(3,7)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22)>;
 
G:=Group( (1,15,21)(2,16,22)(3,23,9)(4,24,10)(5,11,17)(6,12,18)(7,19,13)(8,20,14), (1,15,21)(2,22,16)(3,23,9)(4,10,24)(5,11,17)(6,18,12)(7,19,13)(8,14,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21), (1,5)(3,7)(9,19)(10,24)(11,21)(12,18)(13,23)(14,20)(15,17)(16,22) );
 
G=PermutationGroup([[(1,15,21),(2,16,22),(3,23,9),(4,24,10),(5,11,17),(6,12,18),(7,19,13),(8,20,14)], [(1,15,21),(2,22,16),(3,23,9),(4,10,24),(5,11,17),(6,18,12),(7,19,13),(8,14,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15),(17,24),(18,23),(19,22),(20,21)], [(1,5),(3,7),(9,19),(10,24),(11,21),(12,18),(13,23),(14,20),(15,17),(16,22)]])
 
G:=TransitiveGroup(24,658);
 

Matrix representation of C32⋊D8⋊C2 ►in GL8(ℤ)

01000000
-1-1000000
00-1-10000
00100000
00001000
00000100
00000010
00000001
,
01000000
-1-1000000
00010000
00-1-10000
00001000
00000100
00000010
00000001
,
00100000
00010000
10000000
-1-1000000
00000010
0000000-1
00000-100
0000-1000
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
10000000
-1-1000000
00100000
00-1-10000
00001000
00000100
000000-10
0000000-1

G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;
 

C32⋊D8⋊C2 in GAP, Magma, Sage, TeX 

C_3^2\rtimes D_8\rtimes C_2
 
% in TeX
 
G:=Group("C3^2:D8:C2");
 
// GroupNames label
 
G:=SmallGroup(288,872);
 
// by ID
 
G=gap.SmallGroup(288,872);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,219,100,675,346,80,2693,2028,362,797,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^2=e^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=e*a*e=c*b*c^-1=a^-1,b*d=d*b,e*b*e=b^-1,d*c*d=c^-1,e*c*e=c^5,e*d*e=c^4*d>;
 
// generators/relations
 

Export

Character table of C32⋊D8⋊C2 in TeX

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