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G = C2.C42order 32 = 25

1st central stem extension by C2 of C42

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

Aliases: C2.1C42, C22.7D4, C22.2Q8, C23.12C22, (C2×C4)⋊2C4, C2.1(C4⋊C4), C22.6(C2×C4), (C22×C4).1C2, C2.1(C22⋊C4), SmallGroup(32,2)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2.C42
C1C2C22C23C22×C4 — C2.C42
C1C2 — C2.C42
C1C23 — C2.C42
C1C23 — C2.C42

Generators and relations for C2.C42
 G = < a,b,c | a2=b4=c4=1, cbc-1=ab=ba, ac=ca >

2C4
2C4
2C4
2C4
2C4
2C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4

Character table of C2.C42

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ2111111111-1-1-1-1-1111-1-1-1    linear of order 2
ρ311111111-1-11111-1-1-1-1-1-1    linear of order 2
ρ411111111-11-1-1-1-1-1-1-1111    linear of order 2
ρ51-11-11-11-11-i-ii-ii-11-1i-ii    linear of order 4
ρ611-111-1-1-1-i-1-iii-iii-i11-1    linear of order 4
ρ71-11-11-11-1-1i-ii-ii1-11-ii-i    linear of order 4
ρ811-111-1-1-1i1-iii-i-i-ii-1-11    linear of order 4
ρ911-111-1-1-1-i1i-i-iiii-i-1-11    linear of order 4
ρ101-1-1-111-11ii11-1-1i-i-ii-i-i    linear of order 4
ρ111-1-1-111-11i-i-1-111i-i-i-iii    linear of order 4
ρ121-11-11-11-11ii-ii-i-11-1-ii-i    linear of order 4
ρ131-1-1-111-11-i-i11-1-1-iii-iii    linear of order 4
ρ1411-111-1-1-1i-1i-i-ii-i-ii11-1    linear of order 4
ρ151-11-11-11-1-1-ii-ii-i1-11i-ii    linear of order 4
ρ161-1-1-111-11-ii-1-111-iiii-i-i    linear of order 4
ρ1722-2-2-2-222000000000000    orthogonal lifted from D4
ρ18222-2-22-2-2000000000000    orthogonal lifted from D4
ρ192-222-2-2-22000000000000    orthogonal lifted from D4
ρ202-2-22-222-2000000000000    symplectic lifted from Q8, Schur index 2

Smallest permutation representation of C2.C42
Regular action on 32 points
Generators in S32
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(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)
(1 31 27 11)(2 19 28 8)(3 29 25 9)(4 17 26 6)(5 21 20 13)(7 23 18 15)(10 22 30 14)(12 24 32 16)

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

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

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

C2.C42 is a maximal subgroup of
C22.SD16  C23.31D4  C4×C22⋊C4  C4×C4⋊C4  C23.7Q8  C23.34D4  C428C4  C425C4  C23.8Q8  C23.23D4  C23.63C23  C24.C22  C23.65C23  C23.67C23  C232D4  C23⋊Q8  C23.10D4  C23.78C23  C23.Q8  C23.11D4  C23.81C23  C23.4Q8  C23.83C23  C23.84C23  C23.3A4  C62.D4  C62.Q8  (C6×C12)⋊2C4
 C2p.C42: C424C4  C6.C42  C10.10C42  D10.3Q8  C14.C42  C22.C42  C26.10C42  D26.Q8 ...
C2.C42 is a maximal quotient of
C22.7C42  C23.9D4  C22.C42  M4(2)⋊4C4  C62.D4  C62.Q8  (C6×C12)⋊2C4
 C2p.C42: C4.9C42  C4.10C42  C426C4  C22.4Q16  C4.C42  C6.C42  C10.10C42  D10.3Q8 ...

Matrix representation of C2.C42 in GL4(𝔽5) generated by

1000
0100
0040
0004
,
3000
0100
0040
0001
,
4000
0200
0001
0010
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[3,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0] >;

C2.C42 in GAP, Magma, Sage, TeX

C_2.C_4^2
% in TeX

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

G:=SmallGroup(32,2);
// by ID

G=gap.SmallGroup(32,2);
# by ID

G:=PCGroup([5,-2,2,-2,2,2,40,61,86]);
// Polycyclic

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

Export

Subgroup lattice of C2.C42 in TeX
Character table of C2.C42 in TeX

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