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G = C2×M4(2)  order 32 = 25

Direct product of C2 and M4(2)

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

Aliases: C2×M4(2), C84C22, C4M4(2), C23.3C4, C4.11C23, (C2×C8)⋊6C2, (C2×C4).6C4, C4.10(C2×C4), (C22×C4).6C2, C2.6(C22×C4), (C2×C4).24C22, C22.11(C2×C4), SmallGroup(32,37)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×M4(2)
C1C2C4C2×C4C22×C4 — C2×M4(2)
C1C2 — C2×M4(2)
C1C2×C4 — C2×M4(2)
C1C2C2C4 — C2×M4(2)

Generators and relations for C2×M4(2)
 G = < a,b,c | a2=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C2
2C22
2C22

Character table of C2×M4(2)

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F8G8H
 size 11112211112222222222
ρ111111111111111111111    trivial
ρ21111-1-11111-1-1-1-11111-1-1    linear of order 2
ρ31111-1-11111-1-111-1-1-1-111    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-11-1-11-1-111-11-1-1-11-1111    linear of order 2
ρ61-11-11-1-1-1111-111-11-11-1-1    linear of order 2
ρ71-11-11-1-1-1111-1-1-11-11-111    linear of order 2
ρ81-11-1-11-1-111-11111-11-1-1-1    linear of order 2
ρ9111111-1-1-1-1-1-1i-i-i-iiii-i    linear of order 4
ρ101111-1-1-1-1-1-111-ii-i-iii-ii    linear of order 4
ρ11111111-1-1-1-1-1-1-iiii-i-i-ii    linear of order 4
ρ121111-1-1-1-1-1-111i-iii-i-ii-i    linear of order 4
ρ131-11-1-1111-1-11-1-iii-i-iii-i    linear of order 4
ρ141-11-11-111-1-1-11i-ii-i-ii-ii    linear of order 4
ρ151-11-1-1111-1-11-1i-i-iii-i-ii    linear of order 4
ρ161-11-11-111-1-1-11-ii-iii-ii-i    linear of order 4
ρ1722-2-2002i-2i-2i2i0000000000    complex lifted from M4(2)
ρ1822-2-200-2i2i2i-2i0000000000    complex lifted from M4(2)
ρ192-2-22002i-2i2i-2i0000000000    complex lifted from M4(2)
ρ202-2-2200-2i2i-2i2i0000000000    complex lifted from M4(2)

Permutation representations of C2×M4(2)
On 16 points - transitive group 16T15
Generators in S16
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 11)(3 16)(4 13)(5 10)(6 15)(7 12)(8 9)

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

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

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

G:=TransitiveGroup(16,15);

Matrix representation of C2×M4(2) in GL3(𝔽17) generated by

1600
0160
0016
,
100
001
040
,
1600
010
0016
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,0,4,0,1,0],[16,0,0,0,1,0,0,0,16] >;

C2×M4(2) in GAP, Magma, Sage, TeX

C_2\times M_{4(2})
% in TeX

G:=Group("C2xM4(2)");
// GroupNames label

G:=SmallGroup(32,37);
// by ID

G=gap.SmallGroup(32,37);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,40,181,58]);
// Polycyclic

G:=Group<a,b,c|a^2=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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