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G = M4(2).8C22  order 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, (C2xD4).7C4, C4.46(C2xD4), C4o(C4.D4), (C2xC4).122D4, C4.D4:6C2, (C22xC4).6C4, (C2xC4).3C23, C23.9(C2xC4), C4o(C4.10D4), C4.10D4:6C2, (C2xM4(2)):9C2, C4.32(C22:C4), (C2xD4).45C22, (C2xQ8).39C22, C22.2(C22:C4), C22.10(C22xC4), (C22xC4).33C22, (C2xC4).7(C2xC4), (C2xC4oD4).3C2, C2.16(C2xC22:C4), SmallGroup(64,94)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — M4(2).8C22
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xC4oD4 — M4(2).8C22
Lower central
C1 — C2 — C22 — M4(2).8C22
Upper central
C1 — C4 — C22xC4 — M4(2).8C22
Jennings
C1 — C2 — C2 — C2xC4 — 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, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4.D4, C4.10D4, C2xM4(2), C2xC4oD4, M4(2).8C22
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C2xC22: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  (C22xC8):C4  2+ 1+4.2C4  C4oD4.D4  (C22xQ8):C4  (C2xC42):C4  (C2xC8):D4  M4(2):21D4  C42.9D4  C4oC2wrC4  C2wrC4:C2  C23.(C2xD4)  (C2xD4).135D4  C4:1D4.C4  (C2xD4).137D4  M4(2).24C23  M4(2).25C23  C42.313C23  C42.12C23  C42.13C23  (C4xD5).D4  (C2xD4).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 ...
 (C2xC4p).D4: C23.5C42  (C2xC8).103D4  C42.131D4  M4(2).10C23  (C6xD4).16C4  (D4xC10).29C4  (D4xC14).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  C4xC4.D4  C4xC4.10D4  C42.97D4  (C22xC4).276D4  M4(2):20D4  C42.128D4  (C4xD5).D4  (C2xD4).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 ...
 (C2xD4).D2p: (C23xC4).C4  C42.96D4  (C6xD4).16C4  (D4xC10).29C4  (D4xC14).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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