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G = C22.50C24order 64 = 26

36th central stem extension by C22 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.51C22, C23.21C23, C22.50C24, C2.132- 1+4, C4⋊Q817C2, (C4×Q8)⋊16C2, (C4×D4).13C2, C22⋊Q818C2, C422C27C2, C4.45(C4○D4), C4⋊C4.75C22, (C2×C4).33C23, C4.4D4.8C2, C42⋊C218C2, (C2×D4).71C22, C22⋊C4.7C22, (C2×Q8).65C22, (C22×C4).75C22, C2.29(C2×C4○D4), SmallGroup(64,237)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.50C24
C1C2C22C2×C4C22×C4C42⋊C2 — C22.50C24
C1C22 — C22.50C24
C1C22 — C22.50C24
C1C22 — C22.50C24

Generators and relations for C22.50C24
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=ba=ab, e2=f2=a, dcd-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, ef=fe >

Subgroups: 141 in 106 conjugacy classes, 75 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C2×C4 [×3], C2×C4 [×10], C2×C4 [×4], D4 [×2], Q8 [×6], C23 [×2], C42, C42 [×6], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C422C2 [×4], C4⋊Q8, C22.50C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×4], C24, C2×C4○D4 [×2], 2- 1+4, C22.50C24

Character table of C22.50C24

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S
 size 1111442222222222224444444
ρ11111111111111111111111111    trivial
ρ21111111111-1-111-1-111-1-1-1-1-1-11    linear of order 2
ρ31111-11-11-1-11-1-1-11-11-1-1-111-111    linear of order 2
ρ41111-11-11-1-1-11-1-1-111-111-1-11-11    linear of order 2
ρ51111-11-1-1-11111111-111-1-1-1-11-1    linear of order 2
ρ61111-11-1-1-11-1-111-1-1-11-11111-1-1    linear of order 2
ρ71111111-11-11-1-1-11-1-1-1-11-1-111-1    linear of order 2
ρ81111111-11-1-11-1-1-11-1-11-111-1-1-1    linear of order 2
ρ911111-1-1-1-1-1-11-11-11-11-1-11-1111    linear of order 2
ρ1011111-1-1-1-1-11-1-111-1-1111-11-1-11    linear of order 2
ρ111111-1-11-111-1-11-1-1-1-1-1111-1-111    linear of order 2
ρ121111-1-11-111111-111-1-1-1-1-111-11    linear of order 2
ρ131111-1-1111-1-11-11-1111-11-11-11-1    linear of order 2
ρ141111-1-1111-11-1-111-1111-11-11-1-1    linear of order 2
ρ1511111-1-11-11-1-11-1-1-11-11-1-1111-1    linear of order 2
ρ1611111-1-11-11111-1111-1-111-1-1-1-1    linear of order 2
ρ172-22-20000022i2i-20-2i-2i000000000    complex lifted from C4○D4
ρ182-22-200000-22i-2i20-2i2i000000000    complex lifted from C4○D4
ρ192-2-220022i-20000-2i00-2i2i0000000    complex lifted from C4○D4
ρ202-2-22002-2i-200002i002i-2i0000000    complex lifted from C4○D4
ρ212-22-2000002-2i-2i-202i2i000000000    complex lifted from C4○D4
ρ222-22-200000-2-2i2i202i-2i000000000    complex lifted from C4○D4
ρ232-2-2200-2-2i20000-2i002i2i0000000    complex lifted from C4○D4
ρ242-2-2200-22i200002i00-2i-2i0000000    complex lifted from C4○D4
ρ2544-4-4000000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.50C24
On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)
(1 7)(2 8)(3 5)(4 6)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)
(2 6)(4 8)(9 29)(10 12)(11 31)(14 24)(16 22)(17 25)(18 20)(19 27)(26 28)(30 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 9 5 31)(2 10 6 32)(3 11 7 29)(4 12 8 30)(13 27 23 17)(14 28 24 18)(15 25 21 19)(16 26 22 20)
(1 21 5 15)(2 16 6 22)(3 23 7 13)(4 14 8 24)(9 19 31 25)(10 26 32 20)(11 17 29 27)(12 28 30 18)

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

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

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

C22.50C24 is a maximal subgroup of
C42.189C23  C42.199C23  C42.211C23  C42.213C23  C42.467C23  C42.469C23  C42.46C23  C42.48C23  C42.473C23  C42.477C23  C42.527C23  C42.528C23  C42.72C23  C42.73C23  C42.531C23  C42.532C23  C22.64C25  C22.69C25  C22.82C25  C22.84C25  C22.95C25  C22.99C25  C22.103C25  C22.105C25  C23.144C24  C22.110C25  C22.129C25  C22.130C25  C22.135C25  C22.150C25  C22.151C25  C22.153C25  C22.155C25  C22.157C25
 C2p.2- 1+4: C42.489C23  C42.491C23  C42.63C23  C42.64C23  C42.494C23  C42.497C23  C42.501C23  C42.505C23 ...
C22.50C24 is a maximal quotient of
C23.229C24  C23.236C24  C23.237C24  C23.238C24  C23.255C24  C23.315C24  C23.321C24  C23.346C24  C24.271C23  C23.348C24  C24.286C23  C23.374C24  C24.295C23  C23.385C24  C24.304C23  C23.396C24  C24.308C23  C23.409C24  C23.412C24  C23.414C24  C24.311C23  C24.313C23  C23.424C24  C23.425C24  C24.315C23  C23.428C24  C23.432C24  C23.433C24  C24.332C23  C42.36Q8  C42.37Q8  C23.473C24  C24.341C23  C23.488C24  C23.494C24  C23.496C24  C4223D4  C428Q8  C23.589C24  C23.600C24  C23.615C24  C23.627C24  C23.640C24  C23.645C24  C23.658C24  C23.659C24  C23.662C24  C23.667C24  C23.675C24  C23.676C24  C23.677C24  C23.683C24  C23.686C24  C23.687C24  C23.688C24  C23.689C24  C23.693C24  C23.698C24  C23.700C24  C23.705C24  C23.708C24  C23.710C24
 C42.D2p: C42.166D4  C42.168D4  C42.170D4  C42.177D4  C42.179D4  C42.182D4  C42.183D4  C42.184D4 ...
 C4⋊C4.D2p: C23.244C24  C23.247C24  C24.220C23  C24.267C23  C24.285C23  C23.369C24  C23.392C24  C23.616C24 ...

Matrix representation of C22.50C24 in GL4(𝔽5) generated by

1000
0100
0040
0004
,
4000
0400
0010
0001
,
1000
0400
0010
0004
,
2000
0200
0001
0040
,
0100
1000
0020
0002
,
4000
0400
0020
0003
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,3] >;

C22.50C24 in GAP, Magma, Sage, TeX

C_2^2._{50}C_2^4
% in TeX

G:=Group("C2^2.50C2^4");
// GroupNames label

G:=SmallGroup(64,237);
// by ID

G=gap.SmallGroup(64,237);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,86,297,69]);
// Polycyclic

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

Export

Character table of C22.50C24 in TeX

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