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G = M4(2)⋊C4order 64 = 26

1st semidirect product of M4(2) and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: M4(2)⋊1C4, C23.40D4, C81(C2×C4), C4.Q82C2, (C2×C4).6Q8, C4.4(C2×Q8), C2.D810C2, C4.15(C4⋊C4), (C2×C4).128D4, C4⋊C4.49C22, C2.3(C8⋊C22), C4.27(C22×C4), (C2×C4).70C23, (C2×C8).13C22, C22.50(C2×D4), C22.10(C4⋊C4), C42⋊C2.7C2, C2.3(C8.C22), (C2×M4(2)).1C2, (C22×C4).38C22, C2.14(C2×C4⋊C4), (C2×C4⋊C4).15C2, (C2×C4).26(C2×C4), SmallGroup(64,109)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2)⋊C4
C1C2C22C2×C4C22×C4C2×M4(2) — M4(2)⋊C4
C1C2C4 — M4(2)⋊C4
C1C22C22×C4 — M4(2)⋊C4
C1C2C2C2×C4 — M4(2)⋊C4

Generators and relations for M4(2)⋊C4
 G = < a,b,c | a8=b2=c4=1, bab=a5, cac-1=a-1, bc=cb >

Subgroups: 89 in 59 conjugacy classes, 41 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), M4(2)⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C8⋊C22, C8.C22, M4(2)⋊C4

Character table of M4(2)⋊C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ21111-1-1-111-1-111-1-111-1-11-11    linear of order 2
ρ31111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ41111-1-1-111-11-1-111-1-11-11-11    linear of order 2
ρ51111-1-1-111-1-11-11-11-111-11-1    linear of order 2
ρ6111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-111-11-11-11-11-11-11-1    linear of order 2
ρ81111111111-1-111-1-111-1-1-1-1    linear of order 2
ρ91-11-1-11-1-111-i-i-i-iiiii-1-111    linear of order 4
ρ101-11-1-11-1-111-i-iiiii-i-i11-1-1    linear of order 4
ρ111-11-11-11-11-1-ii-iii-ii-i-111-1    linear of order 4
ρ121-11-11-11-11-1-iii-ii-i-ii1-1-11    linear of order 4
ρ131-11-11-11-11-1i-ii-i-ii-ii-111-1    linear of order 4
ρ141-11-11-11-11-1i-i-ii-iii-i1-1-11    linear of order 4
ρ151-11-1-11-1-111iiii-i-i-i-i-1-111    linear of order 4
ρ161-11-1-11-1-111ii-i-i-i-iii11-1-1    linear of order 4
ρ17222222-2-2-2-2000000000000    orthogonal lifted from D4
ρ182222-2-22-2-22000000000000    orthogonal lifted from D4
ρ192-22-22-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ202-22-2-2222-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ214-4-44000000000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-4000000000000000000    symplectic lifted from C8.C22, Schur index 2

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

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

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

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

M4(2)⋊C4 is a maximal subgroup of
C24.21D4  C4.10D42C4  C4≀C2⋊C4  C429(C2×C4)  M4(2)⋊6D4  M4(2).7D4  M4(2)⋊Q8  C423Q8  C24.100D4  C4○D4.7Q8  C4○D4.8Q8  C4×C8⋊C22  C4×C8.C22  C42.275C23  C42.276C23  M4(2)⋊14D4  M4(2)⋊15D4  M4(2)⋊16D4  M4(2)⋊17D4  C42.219D4  C42.220D4  C42.448D4  C42.449D4  C42.20C23  C42.21C23  C42.22C23  C42.23C23  C24.183D4  C24.116D4  C24.117D4  C24.118D4  (C2×D4).301D4  (C2×D4).302D4  (C2×D4).303D4  (C2×D4).304D4  M4(2)⋊3Q8  M4(2)⋊4Q8  M4(2)⋊5Q8  M4(2)⋊6Q8  C42.57C23  C42.58C23  C42.59C23  C42.60C23  C42.61C23  C42.62C23  C42.63C23  C42.64C23  C42.492C23  C42.493C23  C42.494C23  C42.495C23  C42.496C23  C42.497C23  C42.498C23
 C4p.(C4⋊C4): M4(2).5Q8  M4(2).6Q8  C4⋊C4.225D6  C4⋊C4.232D6  C23.52D12  C20.47(C4⋊C4)  C20.64(C4⋊C4)  C23.47D20 ...
 (C2×C8).D2p: C4.10D43C4  C4.D43C4  M4(2).12D4  M4(2).13D4  M4(2).Q8  M4(2).2Q8  C8⋊(C4×S3)  C8⋊S3⋊C4 ...
M4(2)⋊C4 is a maximal quotient of
M4(2)⋊1C8  C81M4(2)  C42.92D4  C42.21Q8  C24.152D4  C8⋊C42  C24.67D4  C42.24Q8  C42.26Q8  M4(2)⋊1F5
 C23.D4p: C23.37D8  C23.52D12  C23.47D20  C23.47D28 ...
 C4⋊C4.D2p: C24.159D4  C24.71D4  C42.29Q8  C42.30Q8  C42.31Q8  C8⋊(C4×S3)  C8⋊S3⋊C4  C4⋊C4.225D6 ...

Matrix representation of M4(2)⋊C4 in GL6(𝔽17)

0160000
100000
000010
000001
0016200
0016100
,
100000
010000
0016000
0001600
000010
000001
,
550000
5120000
00111400
001600
0000815
000079

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,5,0,0,0,0,5,12,0,0,0,0,0,0,11,1,0,0,0,0,14,6,0,0,0,0,0,0,8,7,0,0,0,0,15,9] >;

M4(2)⋊C4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes C_4
% in TeX

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

G:=SmallGroup(64,109);
// by ID

G=gap.SmallGroup(64,109);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,332,963,117]);
// Polycyclic

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

Export

Character table of M4(2)⋊C4 in TeX

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