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G = C42.3C4order 64 = 26

3rd non-split extension by C42 of C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C42.3C4, (C2×C4).4D4, C4⋊Q8.3C2, (C2×Q8).3C4, C4.10D4.C2, (C2×Q8).2C22, C2.11(C23⋊C4), C22.14(C22⋊C4), (C2×C4).4(C2×C4), SmallGroup(64,37)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.3C4
C1C2C22C2×C4C2×Q8C4⋊Q8 — C42.3C4
C1C2C22C2×C4 — C42.3C4
C1C2C22C2×Q8 — C42.3C4
C1C2C22C2×Q8 — C42.3C4

Generators and relations for C42.3C4
 G = < a,b,c | a4=b4=1, c4=b2, ab=ba, cac-1=a-1b, cbc-1=a2b >

2C2
2C4
2C4
2C4
2C4
2C4
4C4
2C2×C4
2C2×C4
4Q8
4C8
4Q8
4C8
2M4(2)
2C4⋊C4
2M4(2)
2C4⋊C4

Character table of C42.3C4

 class 12A2B4A4B4C4D4E4F8A8B8C8D
 size 1124444488888
ρ11111111111111    trivial
ρ2111-1-1111-1-11-11    linear of order 2
ρ3111-1-1111-11-11-1    linear of order 2
ρ4111111111-1-1-1-1    linear of order 2
ρ511111-11-1-1-i-iii    linear of order 4
ρ6111-1-1-11-11i-i-ii    linear of order 4
ρ7111-1-1-11-11-iii-i    linear of order 4
ρ811111-11-1-1ii-i-i    linear of order 4
ρ9222002-2-200000    orthogonal lifted from D4
ρ1022200-2-2200000    orthogonal lifted from D4
ρ1144-40000000000    orthogonal lifted from C23⋊C4
ρ124-402-200000000    symplectic faithful, Schur index 2
ρ134-40-2200000000    symplectic faithful, Schur index 2

Permutation representations of C42.3C4
On 16 points - transitive group 16T139
Generators in S16
(1 5)(2 16 6 12)(3 7)(4 14 8 10)(9 13)(11 15)
(1 11 5 15)(2 12 6 16)(3 9 7 13)(4 10 8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,139);

C42.3C4 is a maximal subgroup of
(C2×D4).135D4  (C2×D4).137D4  C42.F5  (Q8×C10).C4
 (C2×Q8).D2p: C42.4D4  (C4×C8).C4  (C2×Q8).D4  C8⋊C4.C4  C4⋊Q8.C4  C42.16D4  C42.17D4  Q8≀C2 ...
C42.3C4 is a maximal quotient of
(C2×C42).C4  C423C8  C2.7C2≀C4  C42.F5  (Q8×C10).C4
 (C2×C4).D4p: (C2×C4).D8  (C2×C12).D4  (C2×Q8).D10  (C2×Q8).D14 ...
 (C2×Q8).D2p: (C2×Q8).Q8  C42.3Dic3  C42.3Dic5  C42.3Dic7 ...

Matrix representation of C42.3C4 in GL4(𝔽3) generated by

1000
0020
0100
0001
,
1001
0020
0100
1002
,
0020
1002
0001
0110
G:=sub<GL(4,GF(3))| [1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1],[1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2],[0,1,0,0,0,0,0,1,2,0,0,1,0,2,1,0] >;

C42.3C4 in GAP, Magma, Sage, TeX

C_4^2._3C_4
% in TeX

G:=Group("C4^2.3C4");
// GroupNames label

G:=SmallGroup(64,37);
// by ID

G=gap.SmallGroup(64,37);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,332,158,681,255,117,1444]);
// Polycyclic

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

Export

Subgroup lattice of C42.3C4 in TeX
Character table of C42.3C4 in TeX

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