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G = C4:M4(2)  order 64 = 26

The semidirect product of C4 and M4(2) acting via M4(2)/C2xC4=C2

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

Aliases: C4:2M4(2), C42.11C4, C42.67C22, C4:C8:11C2, (C2xC4).71D4, C4.72(C2xD4), C4.11(C4:C4), (C2xC4).16Q8, C4.21(C2xQ8), C22.7(C4:C4), (C22xC4).15C4, C23.31(C2xC4), (C2xC42).14C2, (C2xC8).45C22, C2.7(C2xM4(2)), (C2xC4).148C23, (C2xM4(2)).13C2, C22.43(C22xC4), (C22xC4).111C22, C2.9(C2xC4:C4), (C2xC4).56(C2xC4), SmallGroup(64,104)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4:M4(2)
C1C2C4C2xC4C22xC4C2xC42 — C4:M4(2)
C1C22 — C4:M4(2)
C1C2xC4 — C4:M4(2)
C1C2C2C2xC4 — C4:M4(2)

Generators and relations for C4:M4(2)
 G = < a,b,c | a4=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >

Subgroups: 81 in 63 conjugacy classes, 45 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, C23, C42, C42, C2xC8, M4(2), C22xC4, C22xC4, C4:C8, C2xC42, C2xM4(2), C4:M4(2)
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C4:C4, M4(2), C22xC4, C2xD4, C2xQ8, C2xC4:C4, C2xM4(2), C4:M4(2)

Character table of C4:M4(2)

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F8G8H
 size 1111221111222222222244444444
ρ11111111111111111111111111111    trivial
ρ21111-1-1111111-1-11-11-1-1-1-111-1-111-1    linear of order 2
ρ31111-1-1111111-1-11-11-1-1-11-1-111-1-11    linear of order 2
ρ411111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111111111-1-1-1-1-11-11-1-111-1-111-1-1    linear of order 2
ρ61111-1-11111-1-111-1-1-1-111-11-11-11-11    linear of order 2
ρ71111111111-1-1-1-1-11-11-1-1-1-111-1-111    linear of order 2
ρ81111-1-11111-1-111-1-1-1-1111-11-11-11-1    linear of order 2
ρ91111-1-1-1-1-1-1-1-1-1-1111111-iii-ii-i-ii    linear of order 4
ρ10111111-1-1-1-1-1-1111-11-1-1-1iiii-i-i-i-i    linear of order 4
ρ111111-1-1-1-1-1-1-1-1-1-1111111i-i-ii-iii-i    linear of order 4
ρ12111111-1-1-1-1-1-1111-11-1-1-1-i-i-i-iiiii    linear of order 4
ρ13111111-1-1-1-111-1-1-1-1-1-111ii-i-i-i-iii    linear of order 4
ρ141111-1-1-1-1-1-11111-11-11-1-1-ii-iii-ii-i    linear of order 4
ρ15111111-1-1-1-111-1-1-1-1-1-111-i-iiiii-i-i    linear of order 4
ρ161111-1-1-1-1-1-11111-11-11-1-1i-ii-i-ii-ii    linear of order 4
ρ172-2-22-22-222-20000020-20000000000    orthogonal lifted from D4
ρ182-2-222-2-222-200000-2020000000000    orthogonal lifted from D4
ρ192-2-22-222-2-2200000-2020000000000    symplectic lifted from Q8, Schur index 2
ρ202-2-222-22-2-220000020-20000000000    symplectic lifted from Q8, Schur index 2
ρ2122-2-2002i-2i2i-2i00-2200002i-2i00000000    complex lifted from M4(2)
ρ2222-2-200-2i2i-2i2i00-220000-2i2i00000000    complex lifted from M4(2)
ρ2322-2-200-2i2i-2i2i002-200002i-2i00000000    complex lifted from M4(2)
ρ242-22-2002i2i-2i-2i2-2002i0-2i00000000000    complex lifted from M4(2)
ρ252-22-200-2i-2i2i2i2-200-2i02i00000000000    complex lifted from M4(2)
ρ262-22-200-2i-2i2i2i-22002i0-2i00000000000    complex lifted from M4(2)
ρ2722-2-2002i-2i2i-2i002-20000-2i2i00000000    complex lifted from M4(2)
ρ282-22-2002i2i-2i-2i-2200-2i02i00000000000    complex lifted from M4(2)

Smallest permutation representation of C4:M4(2)
On 32 points
Generators in S32
(1 24 31 16)(2 9 32 17)(3 18 25 10)(4 11 26 19)(5 20 27 12)(6 13 28 21)(7 22 29 14)(8 15 30 23)
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)

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

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

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

C4:M4(2) is a maximal subgroup of
C42.5Q8  C42.27D4  C42.388D4  C42.389D4  C42.10Q8  C42.30D4  C42.32D4  C42.C8  D4:M4(2)  Q8:M4(2)  D4:4M4(2)  C42.315D4  C42.316D4  C42.52D4  C42.413D4  C42.414D4  C42.78D4  C42.415D4  C42.416D4  C42.79D4  C42.84D4  C42.86D4  C42.87D4  C42.88D4  C42.90D4  C42.91D4  C42.21Q8  C42.96D4  C42.97D4  C42.102D4  C42.322D4  C42.104D4  M4(2).41D4  C42.430D4  C43:C2  C42:8D4  M4(2):12D4  C42.114D4  C42.115D4  C42:16Q8  C42:Q8  M4(2):8Q8  C42.128D4  C42.131D4  C42.32Q8  C42.257C23  C42.674C23  C42.677C23  C42.259C23  C42.678C23  C42.681C23  C42.266C23  D4xM4(2)  C42.286C23  C42.287C23  Q8xM4(2)  C42.698C23  D4:8M4(2)  Q8:7M4(2)  C42.308C23  C42.310C23  C42.211D4  C42.212D4  C42.444D4  C42.445D4  C42.446D4  C42.219D4  C42.220D4  C42.448D4  C42.449D4  C42.299D4  C42.300D4  C42.301D4  C42.302D4  C42.303D4  C42.304D4
 C4p:M4(2): C8:8M4(2)  C8:7M4(2)  C8:1M4(2)  C12:M4(2)  C12:7M4(2)  C20:5M4(2)  C20:13M4(2)  C20:3M4(2) ...
 C4p.(C4:C4): C42.324D4  C42.106D4  Dic3:4M4(2)  Dic5:5M4(2)  C42.14F5  Dic7:4M4(2) ...
C4:M4(2) is a maximal quotient of
C42.43Q8  C23.28C42  C43.7C2  C42.425D4  C42:8C8  C42:9C8  (C2xC8).195D4  C42.14F5
 C4p:M4(2): C8:8M4(2)  C8:7M4(2)  C8:1M4(2)  C12:M4(2)  C12:7M4(2)  C20:5M4(2)  C20:13M4(2)  C20:3M4(2) ...
 (C2xC4p).Q8: C42.27Q8  Dic3:4M4(2)  Dic5:5M4(2)  Dic7:4M4(2) ...

Matrix representation of C4:M4(2) in GL4(F17) generated by

4000
01300
00160
00016
,
0100
4000
0001
00130
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,1,0,0,0,0,0,0,13,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C4:M4(2) in GAP, Magma, Sage, TeX

C_4\rtimes M_4(2)
% in TeX

G:=Group("C4:M4(2)");
// GroupNames label

G:=SmallGroup(64,104);
// by ID

G=gap.SmallGroup(64,104);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,650,88]);
// Polycyclic

G:=Group<a,b,c|a^4=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^5>;
// generators/relations

Export

Character table of C4:M4(2) in TeX

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