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G = M4(2).8C22order 64 = 26

3rd non-split extension by M4(2) of C22 acting via C22/C2=C2

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

Aliases: M4(2).8C22, (C2×D4).7C4, C4.46(C2×D4), C4(C4.D4), (C2×C4).122D4, C4.D46C2, (C22×C4).6C4, (C2×C4).3C23, C23.9(C2×C4), C4(C4.10D4), C4.10D46C2, (C2×M4(2))⋊9C2, C4.32(C22⋊C4), (C2×D4).45C22, (C2×Q8).39C22, C22.2(C22⋊C4), C22.10(C22×C4), (C22×C4).33C22, (C2×C4).7(C2×C4), (C2×C4○D4).3C2, C2.16(C2×C22⋊C4), SmallGroup(64,94)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — M4(2).8C22
C1C2C4C2×C4C22×C4C2×C4○D4 — M4(2).8C22
C1C2C22 — M4(2).8C22
C1C4C22×C4 — M4(2).8C22
C1C2C2C2×C4 — M4(2).8C22

Generators and relations for M4(2).8C22
 G = < a,b,c,d | a8=b2=c2=1, d2=a2, bab=dad-1=a5, cac=ab, bc=cb, dbd-1=a4b, dcd-1=a4bc >

Subgroups: 121 in 75 conjugacy classes, 39 normal (13 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4.D4 [×2], C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, M4(2).8C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, M4(2).8C22

Character table of M4(2).8C22

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H
 size 1122244112224444444444
ρ11111111111111111111111    trivial
ρ211-1-11-1-1-1-11-1111-11-111-1-11    linear of order 2
ρ311-1-11-1-1-1-11-11111-11-1-111-1    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-1-1111-1-11-11-1-1-111-11-11-1    linear of order 2
ρ611111-1-111111-1-111-1-111-1-1    linear of order 2
ρ711111-1-111111-1-1-1-111-1-111    linear of order 2
ρ811-1-1111-1-11-11-1-11-1-11-11-11    linear of order 2
ρ9111111-1-1-1-1-1-11-1i-i-iii-ii-i    linear of order 4
ρ1011-1-11-1111-11-11-1-i-iiiii-i-i    linear of order 4
ρ11111111-1-1-1-1-1-11-1-iii-i-ii-ii    linear of order 4
ρ1211-1-11-1111-11-11-1ii-i-i-i-iii    linear of order 4
ρ1311-1-111-111-11-1-11-i-i-i-iiiii    linear of order 4
ρ1411111-11-1-1-1-1-1-11i-ii-ii-i-ii    linear of order 4
ρ1511-1-111-111-11-1-11iiii-i-i-i-i    linear of order 4
ρ1611111-11-1-1-1-1-1-11-ii-ii-iii-i    linear of order 4
ρ1722-22-20022-2-220000000000    orthogonal lifted from D4
ρ18222-2-200-2-2-2220000000000    orthogonal lifted from D4
ρ19222-2-200222-2-20000000000    orthogonal lifted from D4
ρ2022-22-200-2-222-20000000000    orthogonal lifted from D4
ρ214-400000-4i4i0000000000000    complex faithful
ρ224-4000004i-4i0000000000000    complex faithful

Permutation representations of M4(2).8C22
On 16 points - transitive group 16T71
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(1 11)(2 16)(3 9)(4 14)(5 15)(6 12)(7 13)(8 10)
(1 8 3 2 5 4 7 6)(9 12 11 14 13 16 15 10)

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

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

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

G:=TransitiveGroup(16,71);

On 16 points - transitive group 16T86
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(1 7)(2 4)(3 5)(6 8)(9 15)(10 12)(11 13)(14 16)
(1 10 3 12 5 14 7 16)(2 15 4 9 6 11 8 13)

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

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

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

G:=TransitiveGroup(16,86);

M4(2).8C22 is a maximal subgroup of
C23.3C42  (C22×C8)⋊C4  2+ 1+4.2C4  C4○D4.D4  (C22×Q8)⋊C4  (C2×C42)⋊C4  (C2×C8)⋊D4  M4(2)⋊21D4  C42.9D4  C4○C2≀C4  C2≀C4⋊C2  C23.(C2×D4)  (C2×D4).135D4  C41D4.C4  (C2×D4).137D4  M4(2).24C23  M4(2).25C23  C42.313C23  C42.12C23  C42.13C23  (C4×D5).D4  (C2×D4).9F5
 M4(2).D2p: M4(2).40D4  M4(2).50D4  C42.427D4  M4(2).8D4  M4(2).9D4  M4(2).37D4  M4(2).38D4  M4(2).19D6 ...
 (C2×C4p).D4: C23.5C42  (C2×C8).103D4  C42.131D4  M4(2).10C23  (C6×D4).16C4  (D4×C10).29C4  (D4×C14).16C4 ...
M4(2).8C22 is a maximal quotient of
C42.371D4  C23.8M4(2)  C42.42D4  C23⋊M4(2)  C23⋊C8⋊C2  C42.372D4  C42.66D4  C42.376D4  C42.69D4  C42.72D4  C42.409D4  C42.410D4  C42.78D4  C42.79D4  C42.417D4  C42.418D4  C42.84D4  C42.86D4  C23.15C42  C4×C4.D4  C4×C4.10D4  C42.97D4  (C22×C4).276D4  M4(2)⋊20D4  C42.128D4  (C4×D5).D4  (C2×D4).9F5
 M4(2).D2p: M4(2).45D4  C42.115D4  M4(2).19D6  M4(2).21D6  M4(2).31D6  M4(2).19D10  M4(2).21D10  M4(2).31D10 ...
 (C2×D4).D2p: (C23×C4).C4  C42.96D4  (C6×D4).16C4  (D4×C10).29C4  (D4×C14).16C4 ...

Matrix representation of M4(2).8C22 in GL4(𝔽5) generated by

0100
0020
0001
2000
,
1000
0400
0010
0004
,
4000
0400
0010
0001
,
0300
0040
0003
4000
G:=sub<GL(4,GF(5))| [0,0,0,2,1,0,0,0,0,2,0,0,0,0,1,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,0,0,4,3,0,0,0,0,4,0,0,0,0,3,0] >;

M4(2).8C22 in GAP, Magma, Sage, TeX

M_4(2)._8C_2^2
% in TeX

G:=Group("M4(2).8C2^2");
// GroupNames label

G:=SmallGroup(64,94);
// by ID

G=gap.SmallGroup(64,94);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,158,963,730,88]);
// Polycyclic

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

Export

Character table of M4(2).8C22 in TeX

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