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G = C42.28C22order 64 = 26

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

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

Aliases: C42.28C22, C4⋊Q86C2, C8⋊C410C2, (C2×C4).41D4, Q8⋊C419C2, 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

C1C2×C4 — C42.28C22
C1C2C4C2×C4C2×C8C8⋊C4 — C42.28C22
C1C2C2×C4 — C42.28C22
C1C22C42 — C42.28C22
C1C2C2C2×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 >

8C2
2C4
2C4
4C4
4C4
4C22
4C22
4C4
4C22
2C2×C4
2C2×C4
2C23
2C8
2Q8
2C8
2D4
2D4
2Q8
2C2×C4
4Q8
4Q8
2C22⋊C4
2C4⋊C4
2C22⋊C4
2C2×Q8

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 15)(2 17 26 12)(3 22 27 9)(4 19 28 14)(5 24 29 11)(6 21 30 16)(7 18 31 13)(8 23 32 10)
(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 22)(10 12)(11 20)(13 18)(14 16)(15 24)(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)| (1,20,25,15)(2,17,26,12)(3,22,27,9)(4,19,28,14)(5,24,29,11)(6,21,30,16)(7,18,31,13)(8,23,32,10), (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)(9,22)(10,12)(11,20)(13,18)(14,16)(15,24)(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( (1,20,25,15)(2,17,26,12)(3,22,27,9)(4,19,28,14)(5,24,29,11)(6,21,30,16)(7,18,31,13)(8,23,32,10), (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)(9,22)(10,12)(11,20)(13,18)(14,16)(15,24)(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([(1,20,25,15),(2,17,26,12),(3,22,27,9),(4,19,28,14),(5,24,29,11),(6,21,30,16),(7,18,31,13),(8,23,32,10)], [(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),(9,22),(10,12),(11,20),(13,18),(14,16),(15,24),(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.(C82D4)  (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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