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G = C4×C32⋊C4  order 144 = 24·32

Direct product of C4 and C32⋊C4

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

Aliases: C4×C32⋊C4, C32⋊1C42, (C3×C12)⋊1C4, C3⋊Dic3⋊4C4, (C4×C3⋊S3).8C2, C3⋊S3.5(C2×C4), (C3×C6).3(C2×C4), C2.2(C2×C32⋊C4), (C2×C32⋊C4).6C2, (C2×C3⋊S3).8C22, SmallGroup(144,132)

Series: Derived ►Chief ►Lower central ►Upper central

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

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

C1
C2
9C2
9C2
2C3
2C3
C4
9C4
9C22
9C4
9C4
9C4
9C4
2C6
2C6
6S3
6S3
6S3
6S3
C32
9C2×C4
9C2×C4
9C2×C4
2C12
2C12
6D6
6D6
6Dic3
6Dic3
C3⋊S3
C3×C6
C3⋊S3
9C42
6C4×S3
6C4×S3
C3⋊Dic3
C32⋊C4
C2×C3⋊S3
C3×C12
C32⋊C4
C32⋊C4
C32⋊C4
C4×C3⋊S3
C2×C32⋊C4
C2×C32⋊C4
C4×C32⋊C4

Character table of C4×C32⋊C4

 class 12A2B2C3A3B4A4B4C4D4E4F4G4H4I4J4K4L6A6B12A12B12C12D
 size 119944119999999999444444
ρ1111111111111111111111111    trivial
ρ2111111-1-11-1-1-1-1-1111-111-1-1-1-1    linear of order 2
ρ3111111-1-1-1-11111-1-1-1-111-1-1-1-1    linear of order 2
ρ411111111-11-1-1-1-1-1-1-11111111    linear of order 2
ρ51-1-1111-ii-1-i-iii-i-111i-1-1-ii-ii    linear of order 4
ρ61-1-1111i-i-1ii-i-ii-111-i-1-1i-ii-i    linear of order 4
ρ711-1-11111-i-1ii-i-iii-i-1111111    linear of order 4
ρ81-1-1111i-i1i-iii-i1-1-1-i-1-1i-ii-i    linear of order 4
ρ911-1-111-1-1i1ii-i-i-i-ii111-1-1-1-1    linear of order 4
ρ101-11-111-iiii1-11-1-ii-i-i-1-1-ii-ii    linear of order 4
ρ111-11-111i-i-i-i1-11-1i-iii-1-1i-ii-i    linear of order 4
ρ121-11-111i-ii-i-11-11-ii-ii-1-1i-ii-i    linear of order 4
ρ131-11-111-ii-ii-11-11i-ii-i-1-1-ii-ii    linear of order 4
ρ141-1-1111-ii1-ii-i-ii1-1-1i-1-1-ii-ii    linear of order 4
ρ1511-1-111-1-1-i1-i-iiiii-i111-1-1-1-1    linear of order 4
ρ1611-1-11111i-1-i-iii-i-ii-1111111    linear of order 4
ρ174400-21440000000000-2111-2-2    orthogonal lifted from C32⋊C4
ρ1844001-2-4-400000000001-222-1-1    orthogonal lifted from C2×C32⋊C4
ρ194400-21-4-40000000000-21-1-122    orthogonal lifted from C2×C32⋊C4
ρ2044001-24400000000001-2-2-211    orthogonal lifted from C32⋊C4
ρ214-400-21-4i4i00000000002-1-ii2i-2i    complex faithful
ρ224-4001-24i-4i0000000000-12-2i2ii-i    complex faithful
ρ234-400-214i-4i00000000002-1i-i-2i2i    complex faithful
ρ244-4001-2-4i4i0000000000-122i-2i-ii    complex faithful

Permutation representations of C4×C32⋊C4
►On 24 points - transitive group 24T240
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 23 10)(6 24 11)(7 21 12)(8 22 9)

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

(1 22 3 24)(2 23 4 21)(5 15 12 18)(6 16 9 19)(7 13 10 20)(8 14 11 17)
 
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), (5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22,3,24)(2,23,4,21)(5,15,12,18)(6,16,9,19)(7,13,10,20)(8,14,11,17)>;
 
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), (5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22,3,24)(2,23,4,21)(5,15,12,18)(6,16,9,19)(7,13,10,20)(8,14,11,17) );
 
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)], [(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,22,3,24),(2,23,4,21),(5,15,12,18),(6,16,9,19),(7,13,10,20),(8,14,11,17)]])
 
G:=TransitiveGroup(24,240);
 

C4×C32⋊C4 is a maximal subgroup of
 C32⋊C4≀C2  C32⋊C4⋊C8  C4.4PSU3(𝔽2)  (C3×C24)⋊C4  C32⋊6C4≀C2  C32⋊7C4≀C2  C4⋊F9  C4.4S3≀C2  C32⋊C4⋊Q8  C4⋊S3≀C2  C4.3PSU3(𝔽2)  C4⋊PSU3(𝔽2)  (C6×C12)⋊5C4
C4×C32⋊C4 is a maximal quotient of
 (C3×C24)⋊C4  C32⋊2C8⋊C4  (C6×C12)⋊2C4

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

3000
0300
0030
0003
,
1031
3003
0003
4200
,
0300
3400
3043
0230
,
4004
0002
0412
0020
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[1,3,0,4,0,0,0,2,3,0,0,0,1,3,3,0],[0,3,3,0,3,4,0,2,0,0,4,3,0,0,3,0],[4,0,0,0,0,0,4,0,0,0,1,2,4,2,2,0] >;
 

C4×C32⋊C4 in GAP, Magma, Sage, TeX 

C_4\times C_3^2\rtimes C_4
 
% in TeX
 
G:=Group("C4xC3^2:C4");
 
// GroupNames label
 
G:=SmallGroup(144,132);
 
// by ID
 
G=gap.SmallGroup(144,132);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,55,3364,256,4613,881]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,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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