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G = C42.78C22  order 64 = 26

21st non-split extension by C42 of C22 acting via C22/C2=C2

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

Aliases: C42.78C22, (C4×C8)⋊6C2, (C2×C4).57D4, Q8⋊C4⋊3C2, C42.C2⋊2C2, D4⋊C4.1C2, C4.15(C4○D4), C2.17(C4○D8), C4⋊C4.18C22, (C2×C8).68C22, C4.4D4.5C2, (C2×C4).113C23, (C2×D4).25C22, C22.109(C2×D4), (C2×Q8).21C22, C2.11(C4.4D4), SmallGroup(64,169)

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

Derived series
C1 — C2×C4 — C42.78C22
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C42.78C22
Lower central
C1 — C2 — C2×C4 — C42.78C22
Upper central
C1 — C22 — C42 — C42.78C22
Jennings
C1 — C2 — C2 — C2×C4 — C42.78C22

Generators and relations for C42.78C22
 G = < a,b,c,d | a4=b4=c2=1, d2=b, ab=ba, cac=a-1b2, ad=da, cbc=b-1, bd=db, dcd-1=a2bc >

C1
C2
C2
C2
8C2
C22
C4
C4
2C4
2C4
4C22
4C4
4C4
4C22
4C22
4C4
C2×C4
C2×C4
C2×C4
2C2×C4
2Q8
2Q8
2C8
2C2×C4
2C8
2C2×C4
2D4
2D4
2C23
C42
C4⋊C4
C2×C8
C2×C8
C2×Q8
C4⋊C4
C2×D4
2C22⋊C4
2C22⋊C4
2C4⋊C4
2C4⋊C4
D4⋊C4
C42.C2
C4.4D4
Q8⋊C4
D4⋊C4
Q8⋊C4
C4×C8
C42.78C22

Character table of C42.78C22

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H
 size 1111822222288822222222
ρ11111111111111111111111    trivial
ρ21111-111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-1-11-11-1-11-11-1-1-11-111    linear of order 2
ρ41111-1-1-11-11-1-111-1111-11-1-1    linear of order 2
ρ51111-1111111-1-1-111111111    linear of order 2
ρ611111111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-11-11-11-111-1-1-11-111    linear of order 2
ρ811111-1-11-11-11-1-1-1111-11-1-1    linear of order 2
ρ92222022-2-2-2-200000000000    orthogonal lifted from D4
ρ1022220-2-2-22-2200000000000    orthogonal lifted from D4
ρ112-22-200020-2000002i-2i2i0-2i00    complex lifted from C4○D4
ρ122-22-2000-2020000-2i000-2i02i2i    complex lifted from C4○D4
ρ132-22-200020-200000-2i2i-2i02i00    complex lifted from C4○D4
ρ142-22-2000-20200002i0002i0-2i-2i    complex lifted from C4○D4
ρ152-2-220-2i2i0000000-√2-√-2√-2√-2√2-√-2-√2√2    complex lifted from C4○D8
ρ162-2-220-2i2i0000000√2√-2-√-2-√-2-√2√-2√2-√2    complex lifted from C4○D8
ρ172-2-2202i-2i0000000-√2√-2-√-2-√-2√2√-2-√2√2    complex lifted from C4○D8
ρ1822-2-20000-2i02i000-√-2-√2-√2√2√-2√2√-2-√-2    complex lifted from C4○D8
ρ1922-2-200002i0-2i000√-2-√2-√2√2-√-2√2-√-2√-2    complex lifted from C4○D8
ρ202-2-2202i-2i0000000√2-√-2√-2√-2-√2-√-2√2-√2    complex lifted from C4○D8
ρ2122-2-200002i0-2i000-√-2√2√2-√2√-2-√2√-2-√-2    complex lifted from C4○D8
ρ2222-2-20000-2i02i000√-2√2√2-√2-√-2-√2-√-2√-2    complex lifted from C4○D8

Smallest permutation representation of C42.78C22
►On 32 points
Generators in S32
(1 19 25 15)(2 20 26 16)(3 21 27 9)(4 22 28 10)(5 23 29 11)(6 24 30 12)(7 17 31 13)(8 18 32 14)

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

C42.78C22 is a maximal subgroup of
 (C4×C8)⋊C4
 C42.D2p: C42.355D4  C42.242D4  C42.244D4  C42.308D4  C42.260D4  C42.269D4  C42.270D4  C42.284D4 ...
 C4⋊C4.D2p: C42.366C23  C42.367C23  C42.390C23  C42.391C23  C42.406C23  C42.407C23  C42.408C23  C42.409C23 ...
C42.78C22 is a maximal quotient of
 C42.56Q8  C2.(C8⋊8D4)  C2.(C8⋊7D4)  C42⋊8C4⋊C2  (C2×Q8).109D4  C4⋊C4.Q8
 C42.D2p: C42.433D4  C42.437D4  C42.264D6  C42.213D6  C42.216D6  C42.264D10  C42.213D10  C42.216D10 ...
 C4⋊C4.D2p: C4⋊C4.84D4  C4⋊C4.85D4  C4⋊C4.94D4  (C2×C8).200D6  Q8⋊C4⋊S3  (C8×Dic5)⋊C2  Q8⋊Dic5⋊C2  (C8×Dic7)⋊C2 ...

Matrix representation of C42.78C22 ►in GL4(𝔽17) generated by

01300
13000
0049
00413
,
1000
0100
00162
00161
,
1000
01600
0010
00116
,
0100
1000
00011
00311
G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,4,4,0,0,9,13],[1,0,0,0,0,1,0,0,0,0,16,16,0,0,2,1],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,0,3,0,0,11,11] >;
 

C42.78C22 in GAP, Magma, Sage, TeX 

C_4^2._{78}C_2^2
 
% in TeX
 
G:=Group("C4^2.78C2^2");
 
// GroupNames label
 
G:=SmallGroup(64,169);
 
// by ID
 
G=gap.SmallGroup(64,169);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,121,295,362,50,963,117,1444,88]);
 
// 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,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^2*b*c>;
 
// generators/relations
 

Export

Subgroup lattice of C42.78C22 in TeX
Character table of C42.78C22 in TeX

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