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G = C23.47D4order 64 = 26

18th non-split extension by C23 of D4 acting via D4/C22=C2

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

Aliases: C23.47D4, C22.6SD16, C4.Q811C2, (C2×C4).39D4, C22⋊C8.6C2, C22⋊Q8.5C2, Q8⋊C413C2, C4.32(C4○D4), C4⋊C4.66C22, (C2×C8).39C22, C2.13(C2×SD16), (C2×C4).108C23, C22.104(C2×D4), (C2×Q8).17C22, C2.18(C8.C22), (C22×C4).54C22, C2.14(C22.D4), (C2×C4⋊C4).16C2, SmallGroup(64,164)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.47D4
C1C2C4C2×C4C4⋊C4C2×C4⋊C4 — C23.47D4
C1C2C2×C4 — C23.47D4
C1C22C22×C4 — C23.47D4
C1C2C2C2×C4 — C23.47D4

Generators and relations for C23.47D4
 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=bd3 >

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 C23.47D4

 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
ρ112-22-200-220002i-2i000000    complex lifted from C4○D4
ρ122-22-200-22000-2i2i000000    complex lifted from C4○D4
ρ132-22-2002-2-2i02i00000000    complex lifted from C4○D4
ρ142-22-2002-22i0-2i00000000    complex lifted from C4○D4
ρ152-2-22-22000000000-2--2-2--2    complex lifted from SD16
ρ162-2-222-2000000000-2-2--2--2    complex lifted from SD16
ρ172-2-222-2000000000--2--2-2-2    complex lifted from SD16
ρ182-2-22-22000000000--2-2--2-2    complex lifted from SD16
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C23.47D4
On 32 points
Generators in S32
(2 30)(4 32)(6 26)(8 28)(9 22)(11 24)(13 18)(15 20)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 22 5 18)(2 12 6 16)(3 20 7 24)(4 10 8 14)(9 25 13 29)(11 31 15 27)(17 26 21 30)(19 32 23 28)

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

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

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

C23.47D4 is a maximal subgroup of
C24.115D4  C24.183D4  C24.118D4  (C2×D4).302D4  (C2×D4).304D4  C42.226D4  C42.231D4  C42.232D4  C234SD16  C24.123D4  C24.127D4  C24.128D4  C4.162+ 1+4  C4.192+ 1+4  C42.284D4  C42.288D4  C42.290D4
 C4⋊C4.D2p: C24.14D4  C4⋊C4.12D4  (C2×C4).SD16  C24.15D4  C42.354C23  C42.359C23  C42.424C23  C42.426C23 ...
 C2p.(C2×SD16): C42.222D4  C42.281D4  C23.39D12  C23.34D20  C23.34D28 ...
C23.47D4 is a maximal quotient of
C4.Q810C4  (C2×C4).19Q16  C24.89D4  (C2×C8).170D4  (C2×C4).28D8
 C23.D4p: C23.36D8  C23.39D12  C23.34D20  C23.34D28 ...
 C4⋊C4.D2p: C24.159D4  C24.160D4  C4.68(C4×D4)  C24.85D4  C2.(C83Q8)  D6.1SD16  D6.2SD16  C4⋊C4.231D6 ...

Matrix representation of C23.47D4 in GL4(𝔽17) generated by

1000
161600
0010
0001
,
16000
01600
0010
0001
,
1000
0100
00160
00016
,
4800
131300
00125
001212
,
161500
0100
001610
00101
G:=sub<GL(4,GF(17))| [1,16,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[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],[4,13,0,0,8,13,0,0,0,0,12,12,0,0,5,12],[16,0,0,0,15,1,0,0,0,0,16,10,0,0,10,1] >;

C23.47D4 in GAP, Magma, Sage, TeX

C_2^3._{47}D_4
% in TeX

G:=Group("C2^3.47D4");
// GroupNames label

G:=SmallGroup(64,164);
// by ID

G=gap.SmallGroup(64,164);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,50,1444,376,88]);
// 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*d^3>;
// generators/relations

Export

Subgroup lattice of C23.47D4 in TeX
Character table of C23.47D4 in TeX

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