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G = C4⋊(C32⋊C4)  order 144 = 24·32

The semidirect product of C4 and C32⋊C4 acting via C32⋊C4/C3⋊S3=C2

metabelian, soluble, monomial

Aliases: C4⋊(C32⋊C4), (C3×C12)⋊2C4, C3⋊S3.4D4, C3⋊S3.2Q8, C32⋊3(C4⋊C4), C3⋊Dic3⋊5C4, (C4×C3⋊S3).9C2, (C3×C6).4(C2×C4), C2.5(C2×C32⋊C4), (C2×C32⋊C4).3C2, (C2×C3⋊S3).9C22, SmallGroup(144,133)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C4⋊(C32⋊C4)
Chief series
C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C4⋊(C32⋊C4)
Lower central
C32 — C3×C6 — C4⋊(C32⋊C4)
Upper central
C1 — C2 — C4

Generators and relations for C4⋊(C32⋊C4)
 G = < a,b,c,d | a4=b3=c3=d4=1, ab=ba, ac=ca, dad-1=a-1, dcd-1=bc=cb, dbd-1=b-1c >

C1
C2
9C2
9C2
2C3
2C3
C4
9C4
9C22
18C4
18C4
2C6
2C6
6S3
6S3
6S3
6S3
C32
9C2×C4
9C2×C4
9C2×C4
2C12
2C12
6Dic3
6Dic3
6D6
6D6
C3⋊S3
C3⋊S3
C3×C6
9C4⋊C4
6C4×S3
6C4×S3
C2×C3⋊S3
C3⋊Dic3
C3×C12
2C32⋊C4
2C32⋊C4
C4×C3⋊S3
C2×C32⋊C4
C2×C32⋊C4
C4⋊(C32⋊C4)

Character table of C4⋊(C32⋊C4)

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B12A12B12C12D
 size 11994421818181818444444
ρ1111111111111111111    trivial
ρ2111111-1-111-1-111-1-1-1-1    linear of order 2
ρ31111111-1-1-1-11111111    linear of order 2
ρ4111111-11-1-11-111-1-1-1-1    linear of order 2
ρ511-1-1111i-ii-i-1111111    linear of order 4
ρ611-1-1111-ii-ii-1111111    linear of order 4
ρ711-1-111-1-i-iii111-1-1-1-1    linear of order 4
ρ811-1-111-1ii-i-i111-1-1-1-1    linear of order 4
ρ92-22-222000000-2-20000    orthogonal lifted from D4
ρ102-2-2222000000-2-20000    symplectic lifted from Q8, Schur index 2
ρ114400-214000001-211-2-2    orthogonal lifted from C32⋊C4
ρ1244001-2400000-21-2-211    orthogonal lifted from C32⋊C4
ρ1344001-2-400000-2122-1-1    orthogonal lifted from C2×C32⋊C4
ρ144400-21-4000001-2-1-122    orthogonal lifted from C2×C32⋊C4
ρ154-4001-20000002-1003i-3i    complex faithful
ρ164-400-21000000-123i-3i00    complex faithful
ρ174-4001-20000002-100-3i3i    complex faithful
ρ184-400-21000000-12-3i3i00    complex faithful

Permutation representations of C4⋊(C32⋊C4)
►On 24 points - transitive group 24T238
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(5 21 10)(6 22 11)(7 23 12)(8 24 9)

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

(1 22)(2 21)(3 24)(4 23)(5 20 10 15)(6 19 11 14)(7 18 12 13)(8 17 9 16)
 
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,22)(2,21)(3,24)(4,23)(5,20,10,15)(6,19,11,14)(7,18,12,13)(8,17,9,16)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,22)(2,21)(3,24)(4,23)(5,20,10,15)(6,19,11,14)(7,18,12,13)(8,17,9,16) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(1,22),(2,21),(3,24),(4,23),(5,20,10,15),(6,19,11,14),(7,18,12,13),(8,17,9,16)]])
 
G:=TransitiveGroup(24,238);
 

►On 24 points - transitive group 24T239
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(5 21 10)(6 22 11)(7 23 12)(8 24 9)

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

(1 24 3 22)(2 23 4 21)(5 20 12 13)(6 19 9 16)(7 18 10 15)(8 17 11 14)
 
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,24,3,22)(2,23,4,21)(5,20,12,13)(6,19,9,16)(7,18,10,15)(8,17,11,14)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,24,3,22)(2,23,4,21)(5,20,12,13)(6,19,9,16)(7,18,10,15)(8,17,11,14) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(1,24,3,22),(2,23,4,21),(5,20,12,13),(6,19,9,16),(7,18,10,15),(8,17,11,14)]])
 
G:=TransitiveGroup(24,239);
 

C4⋊(C32⋊C4) is a maximal subgroup of
 C3⋊S3.2D8  C3⋊S3.2Q16  C4.PSU3(𝔽2)  C4.2PSU3(𝔽2)  C8⋊(C32⋊C4)  C3⋊S3.4D8  C3⋊S3.5D8  C3⋊S3.5Q16  S32⋊Q8  S32⋊D4  C4.3PSU3(𝔽2)  C4⋊PSU3(𝔽2)  (C6×C12)⋊5C4  D4×C32⋊C4  Q8×C32⋊C4  C33⋊(C4⋊C4)  C33⋊9(C4⋊C4)
C4⋊(C32⋊C4) is a maximal quotient of
 C8⋊(C32⋊C4)  C3⋊S3.4D8  (C3×C24).C4  C8.(C32⋊C4)  (C3×C12)⋊4C8  C32⋊5(C4⋊C8)  (C6×C12)⋊2C4  C4⋊(He3⋊C4)  C33⋊(C4⋊C4)  C33⋊9(C4⋊C4)

Matrix representation of C4⋊(C32⋊C4) ►in GL4(𝔽5) generated by

3340
2301
1323
3422
,
0342
4232
1143
3340
,
4010
0001
4000
0404
,
4220
3324
4420
4211
G:=sub<GL(4,GF(5))| [3,2,1,3,3,3,3,4,4,0,2,2,0,1,3,2],[0,4,1,3,3,2,1,3,4,3,4,4,2,2,3,0],[4,0,4,0,0,0,0,4,1,0,0,0,0,1,0,4],[4,3,4,4,2,3,4,2,2,2,2,1,0,4,0,1] >;
 

C4⋊(C32⋊C4) in GAP, Magma, Sage, TeX 

C_4\rtimes (C_3^2\rtimes C_4)
 
% in TeX
 
G:=Group("C4:(C3^2:C4)");
 
// GroupNames label
 
G:=SmallGroup(144,133);
 
// by ID
 
G=gap.SmallGroup(144,133);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,121,55,3364,256,4613,881]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
 
// generators/relations
 

Export

Subgroup lattice of C4⋊(C32⋊C4) in TeX
Character table of C4⋊(C32⋊C4) in TeX

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