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G = C42.78C22order 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⋊C43C2, C42.C22C2, 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

C1C2×C4 — C42.78C22
C1C2C4C2×C4C2×C8C4×C8 — C42.78C22
C1C2C2×C4 — C42.78C22
C1C22C42 — C42.78C22
C1C2C2C2×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 >

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

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--2-2-22--2-22    complex lifted from C4○D8
ρ162-2-220-2i2i00000002-2--2--2-2-22-2    complex lifted from C4○D8
ρ172-2-2202i-2i0000000-2-2--2--22-2-22    complex lifted from C4○D8
ρ1822-2-20000-2i02i000--2-2-22-22-2--2    complex lifted from C4○D8
ρ1922-2-200002i0-2i000-2-2-22--22--2-2    complex lifted from C4○D8
ρ202-2-2202i-2i00000002--2-2-2-2--22-2    complex lifted from C4○D8
ρ2122-2-200002i0-2i000--222-2-2-2-2--2    complex lifted from C4○D8
ρ2222-2-20000-2i02i000-222-2--2-2--2-2    complex lifted from C4○D8

Smallest permutation representation of C42.78C22
On 32 points
Generators in S32
(1 19 25 13)(2 20 26 14)(3 21 27 15)(4 22 28 16)(5 23 29 9)(6 24 30 10)(7 17 31 11)(8 18 32 12)
(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 19)(10 12)(11 17)(13 23)(14 16)(15 21)(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)| (1,19,25,13)(2,20,26,14)(3,21,27,15)(4,22,28,16)(5,23,29,9)(6,24,30,10)(7,17,31,11)(8,18,32,12), (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)(9,19)(10,12)(11,17)(13,23)(14,16)(15,21)(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( (1,19,25,13)(2,20,26,14)(3,21,27,15)(4,22,28,16)(5,23,29,9)(6,24,30,10)(7,17,31,11)(8,18,32,12), (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)(9,19)(10,12)(11,17)(13,23)(14,16)(15,21)(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([(1,19,25,13),(2,20,26,14),(3,21,27,15),(4,22,28,16),(5,23,29,9),(6,24,30,10),(7,17,31,11),(8,18,32,12)], [(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),(9,19),(10,12),(11,17),(13,23),(14,16),(15,21),(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.(C88D4)  C2.(C87D4)  C428C4⋊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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