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G = C22.M4(2)  order 64 = 26

2nd non-split extension by C22 of M4(2) acting via M4(2)/C2×C4=C2

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

Aliases: C22.3M4(2), (C2×C4)⋊C8, (C2×C4).92D4, C22⋊C8.1C2, C22.3(C2×C8), (C22×C4).3C4, C2.4(C22⋊C8), C2.2(C23⋊C4), C23.21(C2×C4), (C22×C4).2C22, C2.1(C4.10D4), C22.20(C22⋊C4), (C2×C4⋊C4).1C2, SmallGroup(64,5)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.M4(2)
C1C2C22C2×C4C22×C4C2×C4⋊C4 — C22.M4(2)
C1C2C22 — C22.M4(2)
C1C22C22×C4 — C22.M4(2)
C1C2C22C22×C4 — C22.M4(2)

Generators and relations for C22.M4(2)
 G = < a,b,c,d | a2=b2=c8=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=abc5 >

2C2
2C2
2C22
2C4
2C4
2C22
2C4
2C4
4C4
2C2×C4
2C2×C4
2C2×C4
4C2×C4
4C8
4C2×C4
4C8
2C2×C8
2C2×C8
2C4⋊C4
2C4⋊C4

Character table of C22.M4(2)

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-11-1-111-11    linear of order 2
ρ41111111111-1-1-1-11-111-1-11-1    linear of order 2
ρ5111111-1-1-1-1-11-11i-i-ii-ii-ii    linear of order 4
ρ6111111-1-1-1-1-11-11-iii-ii-ii-i    linear of order 4
ρ7111111-1-1-1-11-11-1-i-ii-i-iiii    linear of order 4
ρ8111111-1-1-1-11-11-1ii-iii-i-i-i    linear of order 4
ρ911-1-1-11-iii-i1i-1-iζ83ζ87ζ85ζ87ζ83ζ8ζ8ζ85    linear of order 8
ρ1011-1-1-11i-i-ii1-i-1iζ85ζ8ζ83ζ8ζ85ζ87ζ87ζ83    linear of order 8
ρ1111-1-1-11-iii-i1i-1-iζ87ζ83ζ8ζ83ζ87ζ85ζ85ζ8    linear of order 8
ρ1211-1-1-11i-i-ii1-i-1iζ8ζ85ζ87ζ85ζ8ζ83ζ83ζ87    linear of order 8
ρ1311-1-1-11i-i-ii-1i1-iζ8ζ8ζ87ζ85ζ85ζ87ζ83ζ83    linear of order 8
ρ1411-1-1-11-iii-i-1-i1iζ83ζ83ζ85ζ87ζ87ζ85ζ8ζ8    linear of order 8
ρ1511-1-1-11-iii-i-1-i1iζ87ζ87ζ8ζ83ζ83ζ8ζ85ζ85    linear of order 8
ρ1611-1-1-11i-i-ii-1i1-iζ85ζ85ζ83ζ8ζ8ζ83ζ87ζ87    linear of order 8
ρ172222-2-2-22-22000000000000    orthogonal lifted from D4
ρ182222-2-22-22-2000000000000    orthogonal lifted from D4
ρ1922-2-22-2-2i-2i2i2i000000000000    complex lifted from M4(2)
ρ2022-2-22-22i2i-2i-2i000000000000    complex lifted from M4(2)
ρ214-44-4000000000000000000    orthogonal lifted from C23⋊C4
ρ224-4-44000000000000000000    symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C22.M4(2)
On 32 points
Generators in S32
(1 5)(2 30)(3 7)(4 32)(6 26)(8 28)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)(25 29)(27 31)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(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 12 25 23)(2 24 26 13)(3 17 27 14)(4 15 28 18)(5 16 29 19)(6 20 30 9)(7 21 31 10)(8 11 32 22)

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

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

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

C22.M4(2) is a maximal subgroup of
C42.371D4  C42.394D4  C42.42D4  C42.44D4  C42.396D4  C42.372D4  (C2×C20)⋊1C8  (C22×C4).F5
 (C2×C4).D4p: C2.C2≀C4  (C2×C4).D8  (C2×C4).5D8  (C2×Dic3)⋊C8  (C2×Dic5)⋊C8  (C2×Dic7)⋊C8 ...
 (C22×C4).D2p: (C2×D4)⋊C8  (C2×C42).C4  C42⋊C8  C423C8  (C2×C4).Q16  C2.7C2≀C4  C23.8M4(2)  (C2×C4)⋊M4(2) ...
C22.M4(2) is a maximal quotient of
C4⋊C4⋊C8  C23.19C42  C42⋊C8  C423C8  C23.2M4(2)  (C2×C20)⋊1C8  (C22×C4).F5
 (C2×C4p).D4: C22.M5(2)  C23.7M4(2)  C42.C8  C22⋊C4.C8  (C2×Dic3)⋊C8  (C2×C12)⋊C8  (C2×Dic5)⋊C8  (C2×C20)⋊C8 ...

Matrix representation of C22.M4(2) in GL6(𝔽17)

100000
010000
001000
000100
0000160
00413016
,
100000
010000
0016000
0001600
0000160
0000016
,
080000
900000
0051296
0031487
00141200
0009215
,
0160000
1600000
00121400
003500
0051296
002098

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,13,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,9,0,0,0,0,8,0,0,0,0,0,0,0,5,3,14,0,0,0,12,14,12,9,0,0,9,8,0,2,0,0,6,7,0,15],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,12,3,5,2,0,0,14,5,12,0,0,0,0,0,9,9,0,0,0,0,6,8] >;

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

C_2^2.M_4(2)
% in TeX

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

G:=SmallGroup(64,5);
// by ID

G=gap.SmallGroup(64,5);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,297,117]);
// Polycyclic

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

Export

Subgroup lattice of C22.M4(2) in TeX
Character table of C22.M4(2) in TeX

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