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G = C4⋊M4(2)  order 64 = 26

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

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

Aliases: C42M4(2), C42.11C4, C42.67C22, C4⋊C811C2, (C2×C4).71D4, C4.72(C2×D4), C4.11(C4⋊C4), (C2×C4).16Q8, C4.21(C2×Q8), C22.7(C4⋊C4), (C22×C4).15C4, C23.31(C2×C4), (C2×C42).14C2, (C2×C8).45C22, C2.7(C2×M4(2)), (C2×C4).148C23, (C2×M4(2)).13C2, C22.43(C22×C4), (C22×C4).111C22, C2.9(C2×C4⋊C4), (C2×C4).56(C2×C4), SmallGroup(64,104)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊M4(2)
C1C2C4C2×C4C22×C4C2×C42 — C4⋊M4(2)
C1C22 — C4⋊M4(2)
C1C2×C4 — C4⋊M4(2)
C1C2C2C2×C4 — 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 [×2], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C4⋊C8 [×4], C2×C42, C2×M4(2) [×2], C4⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C2×M4(2) [×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 14)(2 15 32 17)(3 18 25 16)(4 9 26 19)(5 20 27 10)(6 11 28 21)(7 22 29 12)(8 13 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)| (1,24,31,14)(2,15,32,17)(3,18,25,16)(4,9,26,19)(5,20,27,10)(6,11,28,21)(7,22,29,12)(8,13,30,23), (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( (1,24,31,14)(2,15,32,17)(3,18,25,16)(4,9,26,19)(5,20,27,10)(6,11,28,21)(7,22,29,12)(8,13,30,23), (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([(1,24,31,14),(2,15,32,17),(3,18,25,16),(4,9,26,19),(5,20,27,10),(6,11,28,21),(7,22,29,12),(8,13,30,23)], [(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)  D44M4(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  C428D4  M4(2)⋊12D4  C42.114D4  C42.115D4  C4216Q8  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  D4×M4(2)  C42.286C23  C42.287C23  Q8×M4(2)  C42.698C23  D48M4(2)  Q87M4(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): C88M4(2)  C87M4(2)  C81M4(2)  C12⋊M4(2)  C127M4(2)  C205M4(2)  C2013M4(2)  C203M4(2) ...
 C4p.(C4⋊C4): C42.324D4  C42.106D4  Dic34M4(2)  Dic55M4(2)  C42.14F5  Dic74M4(2) ...
C4⋊M4(2) is a maximal quotient of
C42.43Q8  C23.28C42  C43.7C2  C42.425D4  C428C8  C429C8  (C2×C8).195D4  C42.14F5
 C4p⋊M4(2): C88M4(2)  C87M4(2)  C81M4(2)  C12⋊M4(2)  C127M4(2)  C205M4(2)  C2013M4(2)  C203M4(2) ...
 (C2×C4p).Q8: C42.27Q8  Dic34M4(2)  Dic55M4(2)  Dic74M4(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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