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G = C426C4order 64 = 26

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

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

Aliases: C426C4, C4.1C42, M4(2)⋊2C4, C23.32D4, C4⋊C43C4, 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

C1C4 — C426C4
C1C2C22C23C22×C4C2×C42 — C426C4
C1C2C4 — C426C4
C1C2×C4C22×C4 — C426C4
C1C2C2C22×C4 — C426C4

Generators and relations for C426C4
 G = < a,b,c | a4=b4=c4=1, cac-1=ab=ba, cbc-1=b-1 >

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

Character table of C426C4

 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 C426C4
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 8 6 3)(2 7 5 4)(9 13 11 15)(10 14 12 16)
(1 11 5 14)(2 16 6 9)(3 15 7 12)(4 10 8 13)

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

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,6,3)(2,7,5,4)(9,13,11,15)(10,14,12,16), (1,11,5,14)(2,16,6,9)(3,15,7,12)(4,10,8,13) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,8,6,3),(2,7,5,4),(9,13,11,15),(10,14,12,16)], [(1,11,5,14),(2,16,6,9),(3,15,7,12),(4,10,8,13)])

G:=TransitiveGroup(16,121);

C426C4 is a maximal subgroup of
C24.63D4  C4×C4≀C2  D4.C42  Q8.C42  D4.3C42  C42.102D4  C24.70D4  (C2×C42)⋊C4  C8⋊C417C4  C24.21D4  C4.10D42C4  C4≀C2⋊C4  C429(C2×C4)  M4(2).41D4  M4(2).42D4  C24.72D4  C8.C22⋊C4  C8⋊C22⋊C4  M4(2)⋊19D4  C24.23D4  C4⋊Q815C4  C24.24D4  C4.4D413C4  (C2×C4)≀C2  C427D4  C42.426D4  M4(2).3Q8  M4(2).24D4  C42.427D4  C42.428D4  C42.107D4  C42.62Q8  C42.28Q8  M4(2)⋊7Q8  C4216Q8  C42⋊Q8  C429D4  C42.129D4  C4210D4  C42.130D4  (C2×D4)⋊2Q8  (C2×Q8)⋊2Q8  C42.8D4  M4(2)⋊6D4  M4(2).7D4  M4(2)⋊Q8  C423Q8  C42.32Q8  C23.9S4
 C4p.C42: C8.14C42  C8.5C42  C12.8C42  C12.2C42  C12.3C42  C426Dic5  C20.32C42  C20.33C42 ...
C426C4 is a maximal quotient of
C42.46Q8  C24.46D4  C42.6Q8  C42.7Q8  C24.48D4  C426F5  M4(2)⋊3F5
 C23.D4p: C23.30D8  C12.3C42  C20.33C42  C28.3C42 ...
 C2p.C4≀C2: C426C8  M4(2)⋊C8  C42.4Q8  C42.26D4  C42.388D4  C42.9Q8  C42.370D4  C42.10Q8 ...

Matrix representation of C426C4 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] >;

C426C4 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 C426C4 in TeX
Character table of C426C4 in TeX

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