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G = C42.28C22  order 64 = 26

28th non-split extension by C42 of C22 acting faithfully

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

Aliases: C42.28C22, C4⋊Q8⋊6C2, C8⋊C4⋊10C2, (C2×C4).41D4, Q8⋊C4⋊19C2, D4⋊C4.7C2, C4.16(C4○D4), C4⋊C4.19C22, (C2×C8).54C22, C4.4D4.6C2, C2.20(C8⋊C22), (C2×C4).114C23, (C2×D4).26C22, C22.110(C2×D4), (C2×Q8).22C22, C2.12(C4.4D4), C2.20(C8.C22), SmallGroup(64,170)

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

Derived series
C1 — C2×C4 — C42.28C22
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8⋊C4 — C42.28C22
Lower central
C1 — C2 — C2×C4 — C42.28C22
Upper central
C1 — C22 — C42 — C42.28C22
Jennings
C1 — C2 — C2 — C2×C4 — C42.28C22

Generators and relations for C42.28C22
 G = < a,b,c,d | a4=b4=c2=1, d2=b, ab=ba, cac=a-1b2, dad-1=ab2, cbc=b-1, bd=db, dcd-1=a2b-1c >

C1
C2
C2
C2
8C2
C4
C22
C4
2C4
2C4
4C4
4C4
4C22
4C22
4C4
4C22
C2×C4
C2×C4
C2×C4
2C2×C4
2C2×C4
2C23
2C8
2Q8
2C8
2D4
2D4
2Q8
2C2×C4
4Q8
4Q8
C4⋊C4
C42
C2×C8
C2×Q8
C2×C8
C4⋊C4
C2×D4
2C22⋊C4
2C4⋊C4
2C22⋊C4
2C2×Q8
C4.4D4
Q8⋊C4
D4⋊C4
C4⋊Q8
C8⋊C4
D4⋊C4
Q8⋊C4
C42.28C22

Character table of C42.28C22

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D
 size 1111822448884444
ρ11111111111111111    trivial
ρ21111-111111-11-1-1-1-1    linear of order 2
ρ31111111-1-11-1-11-11-1    linear of order 2
ρ41111-111-1-111-1-11-11    linear of order 2
ρ51111-111-1-1-1111-11-1    linear of order 2
ρ61111111-1-1-1-11-11-11    linear of order 2
ρ71111-11111-1-1-11111    linear of order 2
ρ8111111111-11-1-1-1-1-1    linear of order 2
ρ922220-2-2-220000000    orthogonal lifted from D4
ρ1022220-2-22-20000000    orthogonal lifted from D4
ρ112-22-20-22000002i0-2i0    complex lifted from C4○D4
ρ122-22-202-2000000-2i02i    complex lifted from C4○D4
ρ132-22-202-20000002i0-2i    complex lifted from C4○D4
ρ142-22-20-2200000-2i02i0    complex lifted from C4○D4
ρ154-4-44000000000000    orthogonal lifted from C8⋊C22
ρ1644-4-4000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.28C22
►On 32 points
Generators in S32
(1 20 25 13)(2 17 26 10)(3 22 27 15)(4 19 28 12)(5 24 29 9)(6 21 30 14)(7 18 31 11)(8 23 32 16)

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

(2 28)(3 7)(4 26)(6 32)(8 30)(9 20)(10 16)(11 18)(12 14)(13 24)(15 22)(17 23)(19 21)(27 31)

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

C42.28C22 is a maximal subgroup of
 C42.5C23  C42.8C23  C42.366C23  C42.367C23  C42.385C23  C42.387C23  C42.410C23  C42.411C23  C42.41C23  C42.43C23  C42.46C23  C42.48C23  C42.49C23  C42.51C23  C42.54C23  C42.56C23  C42.508C23  C42.510C23  C42.511C23  C42.512C23  C42.514C23  C42.516C23  C42.517C23  C42.518C23
 C42.D2p: C42.239D4  C42.242D4  C42.243D4  C42.255D4  C42.256D4  C42.257D4  C42.258D4  C42.271D4 ...
 C4p⋊Q8⋊C2: C42.7C23  C42.10C23  C42.425C23  C42.426C23  C12⋊Q8⋊C2  (C2×C8).D6  C20⋊Q8⋊C2  Q8⋊C4⋊D5 ...
C42.28C22 is a maximal quotient of
 C42.24Q8  C2.(C8⋊D4)  C2.(C8⋊2D4)  (C2×C4).24D8  (C2×C4).19Q16  C4⋊C4.Q8
 C42.D2p: C42.110D4  C42.125D4  C42.20D6  C42.62D6  C42.82D6  C42.20D10  C42.62D10  C42.82D10 ...
 C4⋊C4.D2p: (C2×D4)⋊Q8  (C2×Q8)⋊Q8  C4⋊C4.94D4  C12⋊Q8⋊C2  (C2×C8).D6  C20⋊Q8⋊C2  Q8⋊C4⋊D5  C28⋊Q8⋊C2 ...

Matrix representation of C42.28C22 ►in GL6(𝔽17)

1690000
1310000
00831313
0014904
0055127
00012145
,
1600000
0160000
000100
0016000
00441615
0013011
,
100000
4160000
001000
0001600
00441615
0001301
,
400000
040000
0021550
0055127
00831313
0030214

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

C42.28C22 in GAP, Magma, Sage, TeX 

C_4^2._{28}C_2^2
 
% in TeX
 
G:=Group("C4^2.28C2^2");
 
// GroupNames label
 
G:=SmallGroup(64,170);
 
// by ID
 
G=gap.SmallGroup(64,170);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,121,103,362,332,50,963,117,1444,88]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^2=1,d^2=b,a*b=b*a,c*a*c=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c>;
 
// generators/relations
 

Export

Subgroup lattice of C42.28C22 in TeX
Character table of C42.28C22 in TeX

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