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G = C42.C22order 64 = 26

1st non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.1C22, C2.6C4≀C2, C8⋊C46C2, (C2×D4).1C4, (C2×C4).96D4, (C2×Q8).1C4, C4.4D4.1C2, C2.3(C4.D4), C22.37(C22⋊C4), (C2×C4).10(C2×C4), SmallGroup(64,10)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.C22
C1C2C22C2×C4C42C4.4D4 — C42.C22
C1C22C2×C4 — C42.C22
C1C22C42 — C42.C22
C1C22C22C42 — C42.C22

Generators and relations for C42.C22
 G = < a,b,c,d | a4=b4=d2=1, c2=b, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=a2b-1, dcd=a-1b2c >

8C2
2C4
2C4
2C4
4C22
4C4
4C22
4C22
2C8
2C8
2C8
2C8
2C2×C4
2C23
4Q8
4D4
2C2×C8
2C2×C8
2C22⋊C4
2C22⋊C4

Character table of C42.C22

 class 12A2B2C2D4A4B4C4D4E4F8A8B8C8D8E8F8G8H
 size 1111822224844444444
ρ11111111111111111111    trivial
ρ21111-111111-11-1-111-1-11    linear of order 2
ρ311111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-111111-1-111-1-111-1    linear of order 2
ρ511111-1-1-1-11-1ii-i-ii-ii-i    linear of order 4
ρ61111-1-1-1-1-111i-ii-iii-i-i    linear of order 4
ρ711111-1-1-1-11-1-i-iii-ii-ii    linear of order 4
ρ81111-1-1-1-1-111-ii-ii-i-iii    linear of order 4
ρ922220-222-2-2000000000    orthogonal lifted from D4
ρ10222202-2-22-2000000000    orthogonal lifted from D4
ρ112-22-202i00-2i0001-i-1-i001+i-1+i0    complex lifted from C4≀C2
ρ122-22-20-2i002i0001+i-1+i001-i-1-i0    complex lifted from C4≀C2
ρ132-2-2200-2i2i0001+i00-1+i-1-i001-i    complex lifted from C4≀C2
ρ142-2-22002i-2i000-1+i001+i1-i00-1-i    complex lifted from C4≀C2
ρ152-2-22002i-2i0001-i00-1-i-1+i001+i    complex lifted from C4≀C2
ρ162-22-202i00-2i000-1+i1+i00-1-i1-i0    complex lifted from C4≀C2
ρ172-2-2200-2i2i000-1-i001-i1+i00-1+i    complex lifted from C4≀C2
ρ182-22-20-2i002i000-1-i1-i00-1+i1+i0    complex lifted from C4≀C2
ρ1944-4-4000000000000000    orthogonal lifted from C4.D4

Smallest permutation representation of C42.C22
On 32 points
Generators in S32
(1 22 31 15)(2 19 32 12)(3 24 25 9)(4 21 26 14)(5 18 27 11)(6 23 28 16)(7 20 29 13)(8 17 30 10)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(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)
(2 12)(3 29)(4 17)(6 16)(7 25)(8 21)(9 13)(10 26)(11 18)(14 30)(15 22)(19 32)(20 24)(23 28)

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

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

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

C42.C22 is a maximal subgroup of
C42.C23  C42.2C23  C42.3C23  C42.5C23  C42.6C23  C42.7C23  C42.8C23  C42.10C23  (C2×D4).F5  (C2×Q8).F5
 C42.D2p: C42.2D4  C42.3D4  C42.66D4  C42.405D4  C42.407D4  C42.376D4  C42.67D4  C42.69D4 ...
C42.C22 is a maximal quotient of
(C2×C4).98D8  (C2×Q8)⋊C8  C4.C4≀C2  C42.(C2×C4)  (C2×D4).F5  (C2×Q8).F5
 C42.D2p: C42.7Q8  C42.D6  C42.7D6  C42.D10  C42.7D10  C42.D14  C42.7D14 ...

Matrix representation of C42.C22 in GL4(𝔽17) generated by

01300
13000
001615
0011
,
0100
1000
00130
00013
,
11700
71100
001414
00103
,
1000
01600
0010
001616
G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,16,1,0,0,15,1],[0,1,0,0,1,0,0,0,0,0,13,0,0,0,0,13],[11,7,0,0,7,11,0,0,0,0,14,10,0,0,14,3],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;

C42.C22 in GAP, Magma, Sage, TeX

C_4^2.C_2^2
% in TeX

G:=Group("C4^2.C2^2");
// GroupNames label

G:=SmallGroup(64,10);
// by ID

G=gap.SmallGroup(64,10);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,332,158,681,69]);
// Polycyclic

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

Export

Subgroup lattice of C42.C22 in TeX
Character table of C42.C22 in TeX

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