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G = C4.C42order 64 = 26

3rd non-split extension by C4 of C42 acting via C42/C2×C4=C2

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

Aliases: C4.3C42, C23.6Q8, M4(2).2C4, (C2×C8).5C4, (C2×C4).112D4, (C22×C8).3C2, C2.3(C8.C4), C4.20(C22⋊C4), C22.16(C4⋊C4), (C2×M4(2)).7C2, C2.6(C2.C42), (C22×C4).103C22, (C2×C4).64(C2×C4), SmallGroup(64,22)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4.C42
C1C2C4C2×C4C22×C4C22×C8 — C4.C42
C1C2C4 — C4.C42
C1C2×C4C22×C4 — C4.C42
C1C2C2C22×C4 — C4.C42

Generators and relations for C4.C42
 G = < a,b,c | a4=1, b4=c4=a2, bab-1=a-1, ac=ca, cbc-1=a-1b >

2C2
2C2
2C22
2C22
2C8
2C8
2C8
2C8
2C8
2C8
2C2×C8
2C2×C8
2M4(2)
2M4(2)
2C2×C8
2C2×C8

Character table of C4.C42

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F8G8H8I8J8K8L8M8N8O8P
 size 1111221111222222222244444444
ρ11111111111111111111111111111    trivial
ρ211111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-11-1111-1-1-1    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-1-11-1-1-1111    linear of order 2
ρ51111-1-1-1-1-1-1111-1-111-11-1-ii-iiii-i-i    linear of order 4
ρ61111-1-1-1-1-1-1111-1-111-11-1i-ii-i-i-iii    linear of order 4
ρ71-1-11-1111-1-11-1-i-iii-iii-i-i-1ii-i11-1    linear of order 4
ρ81-1-111-1-1-1111-1-ii-ii-i-iii-1-i1-11i-ii    linear of order 4
ρ91-1-111-1-1-1111-1i-ii-iii-i-i-1i1-11-ii-i    linear of order 4
ρ101-1-11-1111-1-11-1ii-i-ii-i-ii-i1ii-i-1-11    linear of order 4
ρ111111-1-1-1-1-1-111-111-1-11-11-i-i-iii-iii    linear of order 4
ρ121-1-11-1111-1-11-1-i-iii-iii-ii1-i-ii-1-11    linear of order 4
ρ131-1-111-1-1-1111-1i-ii-iii-i-i1-i-11-1i-ii    linear of order 4
ρ141-1-11-1111-1-11-1ii-i-ii-i-iii-1-i-ii11-1    linear of order 4
ρ151-1-111-1-1-1111-1-ii-ii-i-iii1i-11-1-ii-i    linear of order 4
ρ161111-1-1-1-1-1-111-111-1-11-11iii-i-ii-i-i    linear of order 4
ρ172-2-22-22-2-222-220000000000000000    orthogonal lifted from D4
ρ182222-2-22222-2-20000000000000000    orthogonal lifted from D4
ρ192-2-222-222-2-2-220000000000000000    orthogonal lifted from D4
ρ20222222-2-2-2-2-2-20000000000000000    symplectic lifted from Q8, Schur index 2
ρ212-22-2002i-2i-2i2i002--2-2-2-2--22-200000000    complex lifted from C8.C4
ρ2222-2-2002i-2i2i-2i00-222-2--2-2--2-200000000    complex lifted from C8.C4
ρ2322-2-200-2i2i-2i2i00-2-2-2-2--22--2200000000    complex lifted from C8.C4
ρ242-22-200-2i2i2i-2i00-2--2-222--2-2-200000000    complex lifted from C8.C4
ρ2522-2-200-2i2i-2i2i00--222--2-2-2-2-200000000    complex lifted from C8.C4
ρ2622-2-2002i-2i2i-2i00--2-2-2--2-22-2200000000    complex lifted from C8.C4
ρ272-22-2002i-2i-2i2i00-2-2--222-2-2--200000000    complex lifted from C8.C4
ρ282-22-200-2i2i2i-2i002-2--2-2-2-22--200000000    complex lifted from C8.C4

Smallest permutation representation of C4.C42
On 32 points
Generators in S32
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(1 17 27 15 5 21 31 11)(2 24 32 10 6 20 28 14)(3 19 29 9 7 23 25 13)(4 18 26 12 8 22 30 16)

G:=sub<Sym(32)| (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,17,27,15,5,21,31,11)(2,24,32,10,6,20,28,14)(3,19,29,9,7,23,25,13)(4,18,26,12,8,22,30,16)>;

G:=Group( (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (1,17,27,15,5,21,31,11)(2,24,32,10,6,20,28,14)(3,19,29,9,7,23,25,13)(4,18,26,12,8,22,30,16) );

G=PermutationGroup([(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(1,17,27,15,5,21,31,11),(2,24,32,10,6,20,28,14),(3,19,29,9,7,23,25,13),(4,18,26,12,8,22,30,16)])

C4.C42 is a maximal subgroup of
C23.12SD16  C23.13SD16  C24.7Q8  Q8.C42  D4.3C42  C24.19Q8  C24.9Q8  C42.322D4  C42.104D4  C24.10Q8  M4(2).40D4  M4(2).43D4  M4(2).44D4  M4(2).48D4  M4(2).49D4  M4(2).3Q8  C4.10D43C4  C4.D43C4  C42.428D4  C42.107D4  C42.430D4  M4(2).4D4  M4(2).5D4  M4(2).6D4  C4⋊C4.96D4  C4⋊C4.97D4  C4⋊C4.98D4  (C2×C8).55D4  (C2×C8).165D4  C42.9D4  M4(2).Q8  M4(2).2Q8  C22⋊C4.Q8
 C4p.C42: C4×C8.C4  C8.6C42  C12.10C42  C12.4C42  C20.40C42  C20.34C42  C20.10C42  M4(2).F5 ...
C4.C42 is a maximal quotient of
M4(2)⋊C8  C42.23D4  C42.25D4  C24.2Q8  C24.3Q8  C42.388D4  C42.370D4  C20.10C42  M4(2).F5
 (C2×C4).D4p: C42.385D4  C42.389D4  C12.10C42  C12.4C42  C20.40C42  C20.34C42  C28.10C42  C28.4C42 ...

Matrix representation of C4.C42 in GL3(𝔽17) generated by

100
040
01513
,
400
0815
009
,
1300
020
0108
G:=sub<GL(3,GF(17))| [1,0,0,0,4,15,0,0,13],[4,0,0,0,8,0,0,15,9],[13,0,0,0,2,10,0,0,8] >;

C4.C42 in GAP, Magma, Sage, TeX

C_4.C_4^2
% in TeX

G:=Group("C4.C4^2");
// GroupNames label

G:=SmallGroup(64,22);
// by ID

G=gap.SmallGroup(64,22);
# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,158,117,1444,88]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^4=c^4=a^2,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b>;
// generators/relations

Export

Subgroup lattice of C4.C42 in TeX
Character table of C4.C42 in TeX

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