Copied to
clipboard

G = C23.48D4order 64 = 26

19th non-split extension by C23 of D4 acting via D4/C22=C2

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

Aliases: C23.48D4, C22.3Q16, C2.D88C2, (C2×C4).40D4, C2.8(C2×Q16), Q8⋊C47C2, C22⋊C8.4C2, (C2×C8).8C22, C22⋊Q8.6C2, C4.33(C4○D4), C4⋊C4.67C22, C2.19(C8⋊C22), (C2×C4).109C23, C22.105(C2×D4), (C2×Q8).18C22, (C22×C4).55C22, C2.15(C22.D4), (C2×C4⋊C4).17C2, SmallGroup(64,165)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.48D4
C1C2C4C2×C4C4⋊C4C2×C4⋊C4 — C23.48D4
C1C2C2×C4 — C23.48D4
C1C22C22×C4 — C23.48D4
C1C2C2C2×C4 — C23.48D4

Generators and relations for C23.48D4
 G = < a,b,c,d,e | a2=b2=c2=1, d4=e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd3 >

2C2
2C2
2C22
2C4
2C22
4C4
4C4
4C4
4C4
2C2×C4
2C2×C4
2C8
2C2×C4
2C8
2C2×C4
2C2×C4
2Q8
2Q8
2C2×C4
4C2×C4
4C2×C4
2C4⋊C4
2C4⋊C4
2C22×C4
2C22⋊C4

Character table of C23.48D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 1111222244444884444
ρ11111111111111111111    trivial
ρ211111111-11-1-1-1-1-11111    linear of order 2
ρ31111-1-111-1-1-111-11-111-1    linear of order 2
ρ41111-1-1111-11-1-11-1-111-1    linear of order 2
ρ51111111111111-1-1-1-1-1-1    linear of order 2
ρ611111111-11-1-1-111-1-1-1-1    linear of order 2
ρ71111-1-111-1-1-1111-11-1-11    linear of order 2
ρ81111-1-1111-11-1-1-111-1-11    linear of order 2
ρ92222-2-2-2-202000000000    orthogonal lifted from D4
ρ10222222-2-20-2000000000    orthogonal lifted from D4
ρ112-2-22-220000000002-22-2    symplectic lifted from Q16, Schur index 2
ρ122-2-222-200000000022-2-2    symplectic lifted from Q16, Schur index 2
ρ132-2-22-22000000000-22-22    symplectic lifted from Q16, Schur index 2
ρ142-2-222-2000000000-2-222    symplectic lifted from Q16, Schur index 2
ρ152-22-200-220002i-2i000000    complex lifted from C4○D4
ρ162-22-200-22000-2i2i000000    complex lifted from C4○D4
ρ172-22-2002-2-2i02i00000000    complex lifted from C4○D4
ρ182-22-2002-22i0-2i00000000    complex lifted from C4○D4
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C23.48D4
On 32 points
Generators in S32
(2 31)(4 25)(6 27)(8 29)(10 20)(12 22)(14 24)(16 18)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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 18 5 22)(2 15 6 11)(3 24 7 20)(4 13 8 9)(10 32 14 28)(12 30 16 26)(17 27 21 31)(19 25 23 29)

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

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

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

C23.48D4 is a maximal subgroup of
C24.115D4  C24.183D4  C24.118D4  (C2×D4).302D4  (C2×D4).303D4  C42.224D4  C42.225D4  C42.228D4  C42.235D4  C24.121D4  C233Q16  C24.126D4  C24.129D4  C4.162+ 1+4  C4.182+ 1+4  C42.284D4  C42.288D4  C42.291D4
 C4⋊C4.D2p: C24.17D4  C4⋊C4.18D4  C4⋊C4.20D4  C24.18D4  C42.354C23  C42.358C23  C42.424C23  C42.425C23 ...
 (C2×C2p).Q16: C42.282D4  C23.40D12  C23.35D20  C23.35D28 ...
C23.48D4 is a maximal quotient of
C24.157D4  C24.88D4
 (C2×C2p).Q16: C2.D85C4  C2.(C4×Q16)  (C2×C4).19Q16  (C2×C8).1Q8  (C2×C8).60D4  (C2×C4).23Q16  C23.40D12  C4⋊C4.230D6 ...
 C4⋊C4.D2p: C23.37D8  C24.160D4  C24.86D4  D6.Q16  D6.2Q16  D10.7Q16  D10.8Q16  D14.Q16 ...

Matrix representation of C23.48D4 in GL4(𝔽17) generated by

1000
01600
0010
0001
,
16000
01600
0010
0001
,
1000
0100
00160
00016
,
0400
13000
001111
0030
,
0100
1000
001010
00127
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,11,3,0,0,11,0],[0,1,0,0,1,0,0,0,0,0,10,12,0,0,10,7] >;

C23.48D4 in GAP, Magma, Sage, TeX

C_2^3._{48}D_4
% in TeX

G:=Group("C2^3.48D4");
// GroupNames label

G:=SmallGroup(64,165);
// by ID

G=gap.SmallGroup(64,165);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,194,1444,376,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^3>;
// generators/relations

Export

Subgroup lattice of C23.48D4 in TeX
Character table of C23.48D4 in TeX

׿
×
𝔽