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G = C42⋊6C4  order 64 = 26

3rd semidirect product of C42 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42⋊6C4, C4.1C42, M4(2)⋊2C4, C23.32D4, C4⋊C4⋊3C4, C2.3C4≀C2, C4.2(C4⋊C4), (C2×C4).11Q8, (C2×C4).141D4, (C2×C42).6C2, C22.3(C4⋊C4), C4.27(C22⋊C4), C42⋊C2.2C2, (C2×M4(2)).6C2, C22.24(C22⋊C4), C2.4(C2.C42), (C22×C4).101C22, (C2×C4).63(C2×C4), SmallGroup(64,20)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C42⋊6C4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊6C4
Lower central
C1 — C2 — C4 — C42⋊6C4
Upper central
C1 — C2×C4 — C22×C4 — C42⋊6C4
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊6C4

Generators and relations for C42⋊6C4
 G = < a,b,c | a4=b4=c4=1, cac-1=ab=ba, cbc-1=b-1 >

C1
C2
C2
C2
2C2
2C2
C22
C4
C22
C22
C4
C4
C4
2C22
2C4
2C4
2C4
2C4
2C22
4C4
4C4
C2×C4
C2×C4
C2×C4
C2×C4
C2×C4
C23
C2×C4
2C2×C4
2C2×C4
2C2×C4
2C8
2C2×C4
2C2×C4
2C8
2C2×C4
2C2×C4
2C2×C4
M4(2)
C42
C42
M4(2)
C4⋊C4
C4⋊C4
C22×C4
2C22×C4
2C42
2C22⋊C4
2C2×C8
2C42
2M4(2)
C2×C42
C42⋊C2
C2×M4(2)
C42⋊6C4

Character table of C42⋊6C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R8A8B8C8D
 size 1111221111222222222244444444
ρ11111111111111111111111111111    trivial
ρ211111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-111-1-1-1-11111-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-111-1-1-1-1-1-1-1-11111    linear of order 2
ρ51-11-1-1111-1-1-i-i-i-i-11iiiii-i-ii1-1-11    linear of order 4
ρ61-11-11-1-1-11111-1-1-11-1-111ii-i-i-i-iii    linear of order 4
ρ71-11-1-1111-1-1iiii-11-i-i-i-ii-i-ii-111-1    linear of order 4
ρ81-11-1-1111-1-1-i-i-i-i-11iiii-iii-i-111-1    linear of order 4
ρ91-11-11-1-1-11111-1-1-11-1-111-i-iiiii-i-i    linear of order 4
ρ101-11-1-1111-1-1iiii-11-i-i-i-i-iii-i1-1-11    linear of order 4
ρ111111-1-1-1-1-1-1ii-i-i11ii-i-i1-11-1-ii-ii    linear of order 4
ρ121111-1-1-1-1-1-1-i-iii11-i-iii1-11-1i-ii-i    linear of order 4
ρ131-11-11-1-1-111-1-111-1111-1-1ii-i-iii-i-i    linear of order 4
ρ141111-1-1-1-1-1-1-i-iii11-i-iii-11-11-ii-ii    linear of order 4
ρ151111-1-1-1-1-1-1ii-i-i11ii-i-i-11-11i-ii-i    linear of order 4
ρ161-11-11-1-1-111-1-111-1111-1-1-i-iii-i-iii    linear of order 4
ρ17222222-2-2-2-20000-2-2000000000000    orthogonal lifted from D4
ρ182222-2-222220000-2-2000000000000    orthogonal lifted from D4
ρ192-22-22-222-2-200002-2000000000000    orthogonal lifted from D4
ρ202-22-2-22-2-22200002-2000000000000    symplectic lifted from Q8, Schur index 2
ρ212-2-2200-2i2i-2i2i1-i-1+i-1-i1+i00-1+i1-i1+i-1-i00000000    complex lifted from C4≀C2
ρ2222-2-2002i-2i-2i2i-1+i1-i-1-i1+i00-1+i1-i-1-i1+i00000000    complex lifted from C4≀C2
ρ232-2-22002i-2i2i-2i1+i-1-i-1+i1-i00-1-i1+i1-i-1+i00000000    complex lifted from C4≀C2
ρ2422-2-200-2i2i2i-2i-1-i1+i-1+i1-i00-1-i1+i-1+i1-i00000000    complex lifted from C4≀C2
ρ252-2-22002i-2i2i-2i-1-i1+i1-i-1+i001+i-1-i-1+i1-i00000000    complex lifted from C4≀C2
ρ2622-2-200-2i2i2i-2i1+i-1-i1-i-1+i001+i-1-i1-i-1+i00000000    complex lifted from C4≀C2
ρ2722-2-2002i-2i-2i2i1-i-1+i1+i-1-i001-i-1+i1+i-1-i00000000    complex lifted from C4≀C2
ρ282-2-2200-2i2i-2i2i-1+i1-i1+i-1-i001-i-1+i-1-i1+i00000000    complex lifted from C4≀C2

Permutation representations of C42⋊6C4
►On 16 points - transitive group 16T121
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

(1 4 7 5)(2 3 8 6)(9 15 11 13)(10 16 12 14)

(1 9 8 14)(2 16 7 11)(3 10 5 15)(4 13 6 12)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,7,5)(2,3,8,6)(9,15,11,13)(10,16,12,14), (1,9,8,14)(2,16,7,11)(3,10,5,15)(4,13,6,12)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,7,5)(2,3,8,6)(9,15,11,13)(10,16,12,14), (1,9,8,14)(2,16,7,11)(3,10,5,15)(4,13,6,12) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,7,5),(2,3,8,6),(9,15,11,13),(10,16,12,14)], [(1,9,8,14),(2,16,7,11),(3,10,5,15),(4,13,6,12)]])
 
G:=TransitiveGroup(16,121);
 

C42⋊6C4 is a maximal subgroup of
 C24.63D4  C4×C4≀C2  D4.C42  Q8.C42  D4.3C42  C42.102D4  C24.70D4  (C2×C42)⋊C4  C8⋊C4⋊17C4  C24.21D4  C4.10D4⋊2C4  C4≀C2⋊C4  C42⋊9(C2×C4)  M4(2).41D4  M4(2).42D4  C24.72D4  C8.C22⋊C4  C8⋊C22⋊C4  M4(2)⋊19D4  C24.23D4  C4⋊Q8⋊15C4  C24.24D4  C4.4D4⋊13C4  (C2×C4)≀C2  C42⋊7D4  C42.426D4  M4(2).3Q8  M4(2).24D4  C42.427D4  C42.428D4  C42.107D4  C42.62Q8  C42.28Q8  M4(2)⋊7Q8  C42⋊16Q8  C42⋊Q8  C42⋊9D4  C42.129D4  C42⋊10D4  C42.130D4  (C2×D4)⋊2Q8  (C2×Q8)⋊2Q8  C42.8D4  M4(2)⋊6D4  M4(2).7D4  M4(2)⋊Q8  C42⋊3Q8  C42.32Q8  C23.9S4
 C4p.C42: C8.14C42  C8.5C42  C12.8C42  C12.2C42  C12.3C42  C42⋊6Dic5  C20.32C42  C20.33C42 ...
C42⋊6C4 is a maximal quotient of
 C42.46Q8  C24.46D4  C42.6Q8  C42.7Q8  C24.48D4  C42⋊6F5  M4(2)⋊3F5
 C23.D4p: C23.30D8  C12.3C42  C20.33C42  C28.3C42 ...
 C2p.C4≀C2: C42⋊6C8  M4(2)⋊C8  C42.4Q8  C42.26D4  C42.388D4  C42.9Q8  C42.370D4  C42.10Q8 ...

Matrix representation of C42⋊6C4 ►in GL3(𝔽17) generated by

100
010
004
,
100
040
0013
,
1300
001
010
G:=sub<GL(3,GF(17))| [1,0,0,0,1,0,0,0,4],[1,0,0,0,4,0,0,0,13],[13,0,0,0,0,1,0,1,0] >;
 

C42⋊6C4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_6C_4
 
% in TeX
 
G:=Group("C4^2:6C4");
 
// GroupNames label
 
G:=SmallGroup(64,20);
 
// by ID
 
G=gap.SmallGroup(64,20);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,158,1444]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=c^4=1,c*a*c^-1=a*b=b*a,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C42⋊6C4 in TeX
Character table of C42⋊6C4 in TeX

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