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G = C4.9C42order 64 = 26

1st central stem extension by C4 of C42

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

Aliases: C421C4, C4.9C42, C23.8D4, (C2×C8)⋊1C4, (C2×C4).8D4, C4.1(C4⋊C4), (C2×C4).1Q8, C22.1(C4⋊C4), C4.17(C22⋊C4), C42⋊C2.1C2, (C2×M4(2)).4C2, C22.6(C22⋊C4), (C22×C4).18C22, C2.2(C2.C42), (C2×C4).61(C2×C4), SmallGroup(64,18)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4.9C42
C1C2C22C23C22×C4C42⋊C2 — C4.9C42
C1C4 — C4.9C42
C1C4 — C4.9C42
C1C2C2C22×C4 — C4.9C42

Generators and relations for C4.9C42
 G = < a,b,c | a4=b4=c4=1, cbc-1=ab=ba, ac=ca >

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

Character table of C4.9C42

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 1122211222444444444444
ρ11111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ311111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ511-11-1111-1-1i-i-iiii-i-i-11-11    linear of order 4
ρ6111-1-111-1-11-ii-ii-11-11-iii-i    linear of order 4
ρ7111-1-111-1-11-ii-ii1-11-1i-i-ii    linear of order 4
ρ811-11-1111-1-1i-i-ii-i-iii1-11-1    linear of order 4
ρ911-1-1111-11-111-1-1i-i-ii-i-iii    linear of order 4
ρ1011-1-1111-11-1-1-111-iii-i-i-iii    linear of order 4
ρ11111-1-111-1-11i-ii-i1-11-1-iii-i    linear of order 4
ρ1211-11-1111-1-1-iii-i-i-iii-11-11    linear of order 4
ρ1311-1-1111-11-111-1-1-iii-iii-i-i    linear of order 4
ρ1411-1-1111-11-1-1-111i-i-iiii-i-i    linear of order 4
ρ1511-11-1111-1-1-iii-iii-i-i1-11-1    linear of order 4
ρ16111-1-111-1-11i-ii-i-11-11i-i-ii    linear of order 4
ρ17222-2-2-2-222-2000000000000    orthogonal lifted from D4
ρ1822-2-22-2-22-22000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ2022-22-2-2-2-222000000000000    symplectic lifted from Q8, Schur index 2
ρ214-40004i-4i000000000000000    complex faithful
ρ224-4000-4i4i000000000000000    complex faithful

Permutation representations of C4.9C42
On 16 points - transitive group 16T74
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5 11 14)(2 6 12 15)(3 7 9 16)(4 8 10 13)
(1 11 3 9)(2 12 4 10)(5 13)(6 14)(7 15)(8 16)

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

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

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

G:=TransitiveGroup(16,74);

On 16 points - transitive group 16T123
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 11)(2 12)(3 9)(4 10)(5 15 7 13)(6 16 8 14)
(1 7)(2 8)(3 5)(4 6)(9 14 11 16)(10 15 12 13)

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

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

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

G:=TransitiveGroup(16,123);

C4.9C42 is a maximal subgroup of
C23.D8  C23.2D8  (C2×D4).24Q8  (C2×C42)⋊C4  C8.(C4⋊C4)  C8⋊C417C4  C42.5D4  C42.6D4  C4.(C4×D4)  (C2×C8)⋊4D4  C42⋊D4  C42.7D4  C422D4  C42.8D4  C22⋊C4.7D4  C42.9D4  (C2×C8).D4  (C2×C8).6D4  C42.10D4  C42.32Q8
 C4p.C42: C8.16C42  C423Dic3  C12.20C42  C421Dic5  C23.9D20  C423F5  C20.24C42  C42⋊Dic7 ...
C4.9C42 is a maximal quotient of
C42.20D4  C24.46D4  C42.4Q8  C42.23D4  C42.25D4  C20.24C42
 C23.D4p: C23.8D8  C12.20C42  C23.9D20  C23.9D28 ...
 (C4×C4p)⋊C4: C421C8  C42.5Q8  C42.6Q8  C423Dic3  C421Dic5  C423F5  C42⋊Dic7 ...

Matrix representation of C4.9C42 in GL4(𝔽5) generated by

2000
0200
0020
0002
,
0002
4000
0200
0010
,
3000
0400
0020
0001
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,4,0,0,0,0,2,0,0,0,0,1,2,0,0,0],[3,0,0,0,0,4,0,0,0,0,2,0,0,0,0,1] >;

C4.9C42 in GAP, Magma, Sage, TeX

C_4._9C_4^2
% in TeX

G:=Group("C4.9C4^2");
// GroupNames label

G:=SmallGroup(64,18);
// by ID

G=gap.SmallGroup(64,18);
# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,158,489,1444]);
// Polycyclic

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

Export

Subgroup lattice of C4.9C42 in TeX
Character table of C4.9C42 in TeX

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