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G = C42.29C22order 64 = 26

29th non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.29C22, C8⋊C411C2, (C2×C4).42D4, C41D4.5C2, C42.C23C2, D4⋊C419C2, C4.17(C4○D4), C4⋊C4.20C22, (C2×C8).55C22, C2.21(C8⋊C22), (C2×C4).115C23, (C2×D4).27C22, C22.111(C2×D4), C2.13(C4.4D4), SmallGroup(64,171)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.29C22
C1C2C4C2×C4C2×C8C8⋊C4 — C42.29C22
C1C2C2×C4 — C42.29C22
C1C22C42 — C42.29C22
C1C2C2C2×C4 — C42.29C22

Generators and relations for C42.29C22
 G = < a,b,c,d | a4=b4=c2=1, d2=b, ab=ba, cac=a-1, dad-1=ab2, cbc=b-1, bd=db, dcd-1=a2b-1c >

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

Character table of C42.29C22

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D
 size 1111882244884444
ρ11111111111111111    trivial
ρ21111-1111-1-11-11-11-1    linear of order 2
ρ311111-111-1-11-1-11-11    linear of order 2
ρ41111-1-1111111-1-1-1-1    linear of order 2
ρ51111111111-1-1-1-1-1-1    linear of order 2
ρ61111-1111-1-1-11-11-11    linear of order 2
ρ711111-111-1-1-111-11-1    linear of order 2
ρ81111-1-11111-1-11111    linear of order 2
ρ9222200-2-2-22000000    orthogonal lifted from D4
ρ10222200-2-22-2000000    orthogonal lifted from D4
ρ112-22-200-220000-2i02i0    complex lifted from C4○D4
ρ122-22-2002-200000-2i02i    complex lifted from C4○D4
ρ132-22-2002-2000002i0-2i    complex lifted from C4○D4
ρ142-22-200-2200002i0-2i0    complex lifted from C4○D4
ρ154-4-44000000000000    orthogonal lifted from C8⋊C22
ρ1644-4-4000000000000    orthogonal lifted from C8⋊C22

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

C42.29C22 is a maximal subgroup of
C4.C4≀C2  C8⋊C45C4
 C42.D2p: C42.239D4  C42.240D4  C42.244D4  C42.255D4  C42.257D4  C42.275D4  C42.277D4  C42.286D4 ...
 C4⋊C4.D2p: C42.2C23  C42.3C23  C42.4C23  C42.366C23  C42.386C23  C42.388C23  C42.406C23  C42.408C23 ...
C42.29C22 is a maximal quotient of
C42.24Q8  C2.(C82D4)  C4⋊C47D4  C428C4⋊C2  (C2×C4).23Q16
 C4⋊C4.D2p: C4⋊C4.84D4  C4⋊C4.D6  C42.70D6  C4⋊C4.D10  C42.70D10  C4⋊C4.D14  C42.70D14 ...
 C42.D2p: C42.112D4  C42.124D4  C42.19D6  C42.72D6  C42.19D10  C42.72D10  C42.19D14  C42.72D14 ...

Matrix representation of C42.29C22 in GL6(𝔽17)

180000
4160000
0006512
0011055
001212011
0051260
,
100000
010000
0001600
001000
0000016
000010
,
100000
4160000
001000
0001600
000001
000010
,
1320000
140000
00125110
001212011
0006512
0011055

G:=sub<GL(6,GF(17))| [1,4,0,0,0,0,8,16,0,0,0,0,0,0,0,11,12,5,0,0,6,0,12,12,0,0,5,5,0,6,0,0,12,5,11,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,4,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[13,1,0,0,0,0,2,4,0,0,0,0,0,0,12,12,0,11,0,0,5,12,6,0,0,0,11,0,5,5,0,0,0,11,12,5] >;

C42.29C22 in GAP, Magma, Sage, TeX

C_4^2._{29}C_2^2
% in TeX

G:=Group("C4^2.29C2^2");
// GroupNames label

G:=SmallGroup(64,171);
// by ID

G=gap.SmallGroup(64,171);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,295,362,332,50,963,117,1444,88]);
// Polycyclic

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

Export

Subgroup lattice of C42.29C22 in TeX
Character table of C42.29C22 in TeX

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