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## G = C2×C33⋊C4order 216 = 23·33

### Direct product of C2 and C33⋊C4

Aliases: C2×C33⋊C4, C6⋊(C32⋊C4), C3⋊S3.5D6, C333(C2×C4), C3⋊S33Dic3, (C32×C6)⋊2C4, (C3×C6)⋊3Dic3, C324(C2×Dic3), (C3×C3⋊S3)⋊5C4, C32(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 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I size 1 1 9 9 2 4 4 4 4 4 4 27 27 27 27 2 4 4 4 4 4 4 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 1 1 1 1 -i -i i i 1 1 1 1 1 1 1 -1 -1 linear of order 4 ρ6 1 -1 1 -1 1 1 1 1 1 1 1 i -i -i i -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 4 ρ7 1 -1 1 -1 1 1 1 1 1 1 1 -i i i -i -1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 1 1 1 1 1 i i -i -i 1 1 1 1 1 1 1 -1 -1 linear of order 4 ρ9 2 2 2 2 -1 -1 -1 -1 2 2 -1 0 0 0 0 -1 2 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 -2 -2 2 -1 -1 -1 -1 2 2 -1 0 0 0 0 1 -2 1 1 -2 1 1 -1 1 orthogonal lifted from D6 ρ11 2 2 -2 -2 -1 -1 -1 -1 2 2 -1 0 0 0 0 -1 2 -1 -1 2 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 2 -2 -1 -1 -1 -1 2 2 -1 0 0 0 0 1 -2 1 1 -2 1 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ13 4 4 0 0 4 1 -2 -2 -2 1 1 0 0 0 0 4 -2 -2 -2 1 1 1 0 0 orthogonal lifted from C32⋊C4 ρ14 4 4 0 0 4 -2 1 1 1 -2 -2 0 0 0 0 4 1 1 1 -2 -2 -2 0 0 orthogonal lifted from C32⋊C4 ρ15 4 -4 0 0 4 1 -2 -2 -2 1 1 0 0 0 0 -4 2 2 2 -1 -1 -1 0 0 orthogonal lifted from C2×C32⋊C4 ρ16 4 -4 0 0 4 -2 1 1 1 -2 -2 0 0 0 0 -4 -1 -1 -1 2 2 2 0 0 orthogonal lifted from C2×C32⋊C4 ρ17 4 4 0 0 -2 1 -1+3√-3/2 -1-3√-3/2 1 -2 1 0 0 0 0 -2 1 -1-3√-3/2 -1+3√-3/2 -2 1 1 0 0 complex lifted from C33⋊C4 ρ18 4 -4 0 0 -2 1 -1-3√-3/2 -1+3√-3/2 1 -2 1 0 0 0 0 2 -1 1-3√-3/2 1+3√-3/2 2 -1 -1 0 0 complex faithful ρ19 4 4 0 0 -2 -1+3√-3/2 1 1 -2 1 -1-3√-3/2 0 0 0 0 -2 -2 1 1 1 -1-3√-3/2 -1+3√-3/2 0 0 complex lifted from C33⋊C4 ρ20 4 -4 0 0 -2 -1-3√-3/2 1 1 -2 1 -1+3√-3/2 0 0 0 0 2 2 -1 -1 -1 1-3√-3/2 1+3√-3/2 0 0 complex faithful ρ21 4 4 0 0 -2 -1-3√-3/2 1 1 -2 1 -1+3√-3/2 0 0 0 0 -2 -2 1 1 1 -1+3√-3/2 -1-3√-3/2 0 0 complex lifted from C33⋊C4 ρ22 4 -4 0 0 -2 -1+3√-3/2 1 1 -2 1 -1-3√-3/2 0 0 0 0 2 2 -1 -1 -1 1+3√-3/2 1-3√-3/2 0 0 complex faithful ρ23 4 4 0 0 -2 1 -1-3√-3/2 -1+3√-3/2 1 -2 1 0 0 0 0 -2 1 -1+3√-3/2 -1-3√-3/2 -2 1 1 0 0 complex lifted from C33⋊C4 ρ24 4 -4 0 0 -2 1 -1+3√-3/2 -1-3√-3/2 1 -2 1 0 0 0 0 2 -1 1+3√-3/2 1-3√-3/2 2 -1 -1 0 0 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)  C339(C4⋊C4)  C6211Dic3  C2×S3×C32⋊C4
C2×C33⋊C4 is a maximal quotient of
C337(C2×C8)  C334M4(2)  C339(C4⋊C4)  C3312M4(2)  C6211Dic3

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

 6 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 5 3 5 3 3 5 2 3 0 0 1 0 0 0 0 4
,
 0 5 2 6 0 2 0 2 3 3 6 1 0 0 0 4
,
 3 6 3 2 6 3 4 2 0 0 2 0 0 0 0 4
,
 0 3 2 0 2 3 4 2 5 2 1 6 2 2 6 3
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

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