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G = C42.6C4order 64 = 26

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

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.6C4, C4.8M4(2), C42.68C22, C22.6M4(2), C4⋊C813C2, C8⋊C47C2, C22⋊C8.7C2, C4.49(C4○D4), (C2×C42).16C2, (C2×C8).47C22, C23.33(C2×C4), (C22×C4).12C4, C2.8(C2×M4(2)), (C2×C4).151C23, C22.45(C22×C4), (C22×C4).94C22, C2.11(C42⋊C2), (C2×C4).58(C2×C4), SmallGroup(64,113)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.6C4
C1C2C4C2×C4C42C2×C42 — C42.6C4
C1C22 — C42.6C4
C1C2×C4 — C42.6C4
C1C2C2C2×C4 — C42.6C4

Generators and relations for C42.6C4
 G = < a,b,c | a4=b4=1, c4=a2, ab=ba, cac-1=a-1b2, cbc-1=a2b >

2C2
2C2
2C4
2C4
2C22
2C4
2C4
2C22
2C8
2C8
2C8
2C2×C4
2C2×C4
2C2×C4
2C8
2C2×C4
2C2×C4
2C2×C4

Character table of C42.6C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F8G8H
 size 1111221111222222222244444444
ρ11111111111111111111111111111    trivial
ρ21111-1-11111-1-1-11-1111-1-1-1-111-1-111    linear of order 2
ρ31111-1-11111-1-1-11-1111-1-111-1-111-1-1    linear of order 2
ρ411111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111111111-1-1-1-1-1-1-1-111-11-11-11-11    linear of order 2
ρ61111-1-11111111-11-1-1-1-1-11-1-111-1-11    linear of order 2
ρ71111111111-1-1-1-1-1-1-1-1111-11-11-11-1    linear of order 2
ρ81111-1-11111111-11-1-1-1-1-1-111-1-111-1    linear of order 2
ρ91111-1-1-1-1-1-111-11-1-11-111-i-iiiii-i-i    linear of order 4
ρ10111111-1-1-1-1-1-1111-11-1-1-1iiii-i-i-i-i    linear of order 4
ρ111111-1-1-1-1-1-111-11-1-11-111ii-i-i-i-iii    linear of order 4
ρ12111111-1-1-1-1-1-1111-11-1-1-1-i-i-i-iiiii    linear of order 4
ρ13111111-1-1-1-111-1-1-11-11-1-1-ii-iii-ii-i    linear of order 4
ρ141111-1-1-1-1-1-1-1-11-111-1111i-i-ii-iii-i    linear of order 4
ρ15111111-1-1-1-111-1-1-11-11-1-1i-ii-i-ii-ii    linear of order 4
ρ161111-1-1-1-1-1-1-1-11-111-1111-iii-ii-i-ii    linear of order 4
ρ172-22-22-22i-2i2i-2i00000000-2i2i00000000    complex lifted from M4(2)
ρ182-2-2200-2i-2i2i2i2-22i0-2i0000000000000    complex lifted from M4(2)
ρ192-2-22002i2i-2i-2i-222i0-2i0000000000000    complex lifted from M4(2)
ρ202-22-22-2-2i2i-2i2i000000002i-2i00000000    complex lifted from M4(2)
ρ2122-2-2002-2-22000-2i02i2i-2i0000000000    complex lifted from C4○D4
ρ222-2-2200-2i-2i2i2i-22-2i02i0000000000000    complex lifted from M4(2)
ρ2322-2-200-222-2000-2i0-2i2i2i0000000000    complex lifted from C4○D4
ρ242-22-2-22-2i2i-2i2i00000000-2i2i00000000    complex lifted from M4(2)
ρ252-22-2-222i-2i2i-2i000000002i-2i00000000    complex lifted from M4(2)
ρ2622-2-200-222-20002i02i-2i-2i0000000000    complex lifted from C4○D4
ρ272-2-22002i2i-2i-2i2-2-2i02i0000000000000    complex lifted from M4(2)
ρ2822-2-2002-2-220002i0-2i-2i2i0000000000    complex lifted from C4○D4

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

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

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

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

C42.6C4 is a maximal subgroup of
C42.677C23  C42.261C23  C42.262C23  C42.678C23  C42.291C23  C42.292C23  D46M4(2)  Q86M4(2)  C233M4(2)  D47M4(2)  C42.300C23  C42.301C23  Q8.4M4(2)  C42.696C23  C42.305C23  C42.698C23  D48M4(2)  C42.307C23  C42.310C23  C42.15F5  C42.7F5
 C42.D2p: C42.4Q8  C42.23D4  C42.6Q8  C42.25D4  C42.26D4  C42.9Q8  C42.370D4  C42.30D4 ...
 (C2×C2p).M4(2): C42.302C23  Dic3.M4(2)  Dic5.9M4(2)  Dic5.13M4(2)  C20.30M4(2)  Dic7.M4(2) ...
C42.6C4 is a maximal quotient of
C43.C2  C23.32M4(2)  C425C8  C8.5M4(2)  C8.19M4(2)  C42.15F5  C42.7F5  Dic5.13M4(2)  C20.30M4(2)
 C42.D2p: C42.378D4  C43.7C2  C42.425D4  C428C8  C42.182D6  C42.202D6  C42.270D6  C42.182D10 ...
 (C2×C8).D2p: (C2×C8).Q8  C23.9M4(2)  Dic3.M4(2)  Dic5.9M4(2)  Dic7.M4(2) ...

Matrix representation of C42.6C4 in GL4(𝔽17) generated by

4000
0400
0040
00013
,
13000
0400
0010
00016
,
0100
4000
0001
0040
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,16],[0,4,0,0,1,0,0,0,0,0,0,4,0,0,1,0] >;

C42.6C4 in GAP, Magma, Sage, TeX

C_4^2._6C_4
% in TeX

G:=Group("C4^2.6C4");
// GroupNames label

G:=SmallGroup(64,113);
// by ID

G=gap.SmallGroup(64,113);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,50,88]);
// Polycyclic

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

Export

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

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