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G = C2×C33⋊C4  order 216 = 23·33

Direct product of C2 and C33⋊C4

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×C33⋊C4, C6⋊(C32⋊C4), C3⋊S3.5D6, C33⋊3(C2×C4), C3⋊S3⋊3Dic3, (C32×C6)⋊2C4, (C3×C6)⋊3Dic3, C32⋊4(C2×Dic3), (C3×C3⋊S3)⋊5C4, C3⋊2(C2×C32⋊C4), (C2×C3⋊S3).3S3, (C6×C3⋊S3).5C2, (C3×C3⋊S3).8C22, SmallGroup(216,169)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C2×C33⋊C4
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C33⋊C4 — C2×C33⋊C4
Lower central
C33 — C2×C33⋊C4
Upper central
C1 — C2

Generators and relations for C2×C33⋊C4
 G = < a,b,c,d,e | a2=b3=c3=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc-1, cd=dc, ece-1=b-1c-1, ede-1=d-1 >

Subgroups: 308 in 60 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C33, C32⋊C4, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C32×C6, C2×C32⋊C4, C33⋊C4, C6×C3⋊S3, C2×C33⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C2×Dic3, C32⋊C4, C2×C32⋊C4, C33⋊C4, C2×C33⋊C4

Character table of C2×C33⋊C4

 class 12A2B2C3A3B3C3D3E3F3G4A4B4C4D6A6B6C6D6E6F6G6H6I
 size 119924444442727272724444441818
ρ1111111111111111111111111    trivial
ρ21-1-111111111-11-11-1-1-1-1-1-1-11-1    linear of order 2
ρ31-1-1111111111-11-1-1-1-1-1-1-1-11-1    linear of order 2
ρ411111111111-1-1-1-1111111111    linear of order 2
ρ511-1-11111111-i-iii1111111-1-1    linear of order 4
ρ61-11-11111111i-i-ii-1-1-1-1-1-1-1-11    linear of order 4
ρ71-11-11111111-iii-i-1-1-1-1-1-1-1-11    linear of order 4
ρ811-1-11111111ii-i-i1111111-1-1    linear of order 4
ρ92222-1-1-1-122-10000-12-1-12-1-1-1-1    orthogonal lifted from S3
ρ102-2-22-1-1-1-122-100001-211-211-11    orthogonal lifted from D6
ρ1122-2-2-1-1-1-122-10000-12-1-12-1-111    symplectic lifted from Dic3, Schur index 2
ρ122-22-2-1-1-1-122-100001-211-2111-1    symplectic lifted from Dic3, Schur index 2
ρ13440041-2-2-21100004-2-2-211100    orthogonal lifted from C32⋊C4
ρ1444004-2111-2-200004111-2-2-200    orthogonal lifted from C32⋊C4
ρ154-40041-2-2-2110000-4222-1-1-100    orthogonal lifted from C2×C32⋊C4
ρ164-4004-2111-2-20000-4-1-1-122200    orthogonal lifted from C2×C32⋊C4
ρ174400-21-1+3√-3/2-1-3√-3/21-210000-21-1-3√-3/2-1+3√-3/2-21100    complex lifted from C33⋊C4
ρ184-400-21-1-3√-3/2-1+3√-3/21-2100002-11-3√-3/21+3√-3/22-1-100    complex faithful
ρ194400-2-1+3√-3/211-21-1-3√-3/20000-2-2111-1-3√-3/2-1+3√-3/200    complex lifted from C33⋊C4
ρ204-400-2-1-3√-3/211-21-1+3√-3/2000022-1-1-11-3√-3/21+3√-3/200    complex faithful
ρ214400-2-1-3√-3/211-21-1+3√-3/20000-2-2111-1+3√-3/2-1-3√-3/200    complex lifted from C33⋊C4
ρ224-400-2-1+3√-3/211-21-1-3√-3/2000022-1-1-11+3√-3/21-3√-3/200    complex faithful
ρ234400-21-1-3√-3/2-1+3√-3/21-210000-21-1+3√-3/2-1-3√-3/2-21100    complex lifted from C33⋊C4
ρ244-400-21-1+3√-3/2-1-3√-3/21-2100002-11+3√-3/21-3√-3/22-1-100    complex faithful

Permutation representations of C2×C33⋊C4
►On 24 points - transitive group 24T553
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 19)(10 20)(11 17)(12 18)(13 24)(14 21)(15 22)(16 23)

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

(2 17 22)(4 24 19)(6 11 15)(8 13 9)

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

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
 
G:=sub<Sym(24)| (1,5)(2,6)(3,7)(4,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,21,20)(2,22,17)(3,18,23)(4,19,24)(5,14,10)(6,15,11)(7,12,16)(8,9,13), (2,17,22)(4,24,19)(6,11,15)(8,13,9), (1,21,20)(2,17,22)(3,23,18)(4,19,24)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;
 
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,19)(10,20)(11,17)(12,18)(13,24)(14,21)(15,22)(16,23), (1,21,20)(2,22,17)(3,18,23)(4,19,24)(5,14,10)(6,15,11)(7,12,16)(8,9,13), (2,17,22)(4,24,19)(6,11,15)(8,13,9), (1,21,20)(2,17,22)(3,23,18)(4,19,24)(5,14,10)(6,11,15)(7,16,12)(8,9,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );
 
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,19),(10,20),(11,17),(12,18),(13,24),(14,21),(15,22),(16,23)], [(1,21,20),(2,22,17),(3,18,23),(4,19,24),(5,14,10),(6,15,11),(7,12,16),(8,9,13)], [(2,17,22),(4,24,19),(6,11,15),(8,13,9)], [(1,21,20),(2,17,22),(3,23,18),(4,19,24),(5,14,10),(6,11,15),(7,16,12),(8,9,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)]])
 
G:=TransitiveGroup(24,553);
 

C2×C33⋊C4 is a maximal subgroup of
 Dic3×C32⋊C4  D6⋊(C32⋊C4)  C33⋊(C4⋊C4)  C3⋊S3.2D12  S32⋊Dic3  C33⋊C4⋊C4  C6.PSU3(𝔽2)  C6.2PSU3(𝔽2)  C33⋊9(C4⋊C4)  C62⋊11Dic3  C2×S3×C32⋊C4
C2×C33⋊C4 is a maximal quotient of
 C33⋊7(C2×C8)  C33⋊4M4(2)  C33⋊9(C4⋊C4)  C33⋊12M4(2)  C62⋊11Dic3

Matrix representation of C2×C33⋊C4 ►in GL4(𝔽7) generated by

6000
0600
0060
0006
,
5353
3523
0010
0004
,
0526
0202
3361
0004
,
3632
6342
0020
0004
,
0320
2342
5216
2263
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[5,3,0,0,3,5,0,0,5,2,1,0,3,3,0,4],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[0,2,5,2,3,3,2,2,2,4,1,6,0,2,6,3] >;
 

C2×C33⋊C4 in GAP, Magma, Sage, TeX 

C_2\times C_3^3\rtimes C_4
 
% in TeX
 
G:=Group("C2xC3^3:C4");
 
// GroupNames label
 
G:=SmallGroup(216,169);
 
// by ID
 
G=gap.SmallGroup(216,169);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,3,-3,24,963,111,964,376,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^-1,c*d=d*c,e*c*e^-1=b^-1*c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Character table of C2×C33⋊C4 in TeX

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