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G = C22.47C24order 64 = 26

33rd central stem extension by C22 of C24

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

Aliases: C22.47C24, C42.48C22, C23.19C23, C2.162+ 1+4, (C4×D4)⋊20C2, C4⋊D415C2, C42.C29C2, C422C26C2, C4.36(C4○D4), C4⋊C4.36C22, (C2×C4).57C23, C42⋊C216C2, (C2×D4).35C22, C22.11(C4○D4), C22.D411C2, C22⋊C4.22C22, (C22×C4).14C22, (C2×C4⋊C4)⋊22C2, C2.26(C2×C4○D4), SmallGroup(64,234)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.47C24
C1C2C22C2×C4C22×C4C2×C4⋊C4 — C22.47C24
C1C22 — C22.47C24
C1C22 — C22.47C24
C1C22 — C22.47C24

Generators and relations for C22.47C24
 G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=ba=ab, f2=a, dcd-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, ef=fe >

Subgroups: 181 in 119 conjugacy classes, 75 normal (31 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C2×C4 [×5], C2×C4 [×6], C2×C4 [×8], D4 [×10], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22.D4 [×2], C42.C2, C422C2 [×2], C22.47C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×4], C24, C2×C4○D4 [×2], 2+ 1+4, C22.47C24

Character table of C22.47C24

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111224442222222222444444
ρ11111111111111111111111111    trivial
ρ21111-1-1-1-1-111-111-1-1-1-1-11-11111    linear of order 2
ρ3111111-111-1-1-1-1-1-1-11-111-1-111-1    linear of order 2
ρ41111-1-11-1-1-1-11-1-111-11-111-111-1    linear of order 2
ρ51111111-11-1-1-1-1-111-1-1-1-1-111-11    linear of order 2
ρ61111-1-1-11-1-1-11-1-1-1-1111-1111-11    linear of order 2
ρ7111111-1-1111111-1-1-11-1-11-11-1-1    linear of order 2
ρ81111-1-111-111-111111-11-1-1-11-1-1    linear of order 2
ρ9111111-11-11-111-111-11-11-11-1-1-1    linear of order 2
ρ101111-1-11-111-1-11-1-1-11-11111-1-1-1    linear of order 2
ρ1111111111-1-11-1-11-1-1-1-1-111-1-1-11    linear of order 2
ρ121111-1-1-1-11-111-11111111-1-1-1-11    linear of order 2
ρ13111111-1-1-1-11-1-11111-11-111-11-1    linear of order 2
ρ141111-1-1111-111-11-1-1-11-1-1-11-11-1    linear of order 2
ρ151111111-1-11-111-1-1-1111-1-1-1-111    linear of order 2
ρ161111-1-1-1111-1-11-111-1-1-1-11-1-111    linear of order 2
ρ172-22-2000000-2002-2i2i-2i02i000000    complex lifted from C4○D4
ρ1822-2-22-20002i02i-2i0000-2i0000000    complex lifted from C4○D4
ρ192-22-2000000200-22i-2i-2i02i000000    complex lifted from C4○D4
ρ2022-2-2-220002i0-2i-2i00002i0000000    complex lifted from C4○D4
ρ212-22-2000000-20022i-2i2i0-2i000000    complex lifted from C4○D4
ρ2222-2-22-2000-2i0-2i2i00002i0000000    complex lifted from C4○D4
ρ232-22-2000000200-2-2i2i2i0-2i000000    complex lifted from C4○D4
ρ2422-2-2-22000-2i02i2i0000-2i0000000    complex lifted from C4○D4
ρ254-4-44000000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of C22.47C24
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)
(1 30)(2 9)(3 32)(4 11)(5 12)(6 31)(7 10)(8 29)(13 26)(14 17)(15 28)(16 19)(18 21)(20 23)(22 25)(24 27)
(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 31)(2 32)(3 29)(4 30)(5 9)(6 10)(7 11)(8 12)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)
(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), (1,30)(2,9)(3,32)(4,11)(5,12)(6,31)(7,10)(8,29)(13,26)(14,17)(15,28)(16,19)(18,21)(20,23)(22,25)(24,27), (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), (1,31)(2,32)(3,29)(4,30)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28), (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), (1,30)(2,9)(3,32)(4,11)(5,12)(6,31)(7,10)(8,29)(13,26)(14,17)(15,28)(16,19)(18,21)(20,23)(22,25)(24,27), (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), (1,31)(2,32)(3,29)(4,30)(5,9)(6,10)(7,11)(8,12)(13,17)(14,18)(15,19)(16,20)(21,25)(22,26)(23,27)(24,28), (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)], [(1,30),(2,9),(3,32),(4,11),(5,12),(6,31),(7,10),(8,29),(13,26),(14,17),(15,28),(16,19),(18,21),(20,23),(22,25),(24,27)], [(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)], [(1,31),(2,32),(3,29),(4,30),(5,9),(6,10),(7,11),(8,12),(13,17),(14,18),(15,19),(16,20),(21,25),(22,26),(23,27),(24,28)], [(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)])

C22.47C24 is a maximal subgroup of
C42.486C23  C42.488C23  C42.59C23  C42.60C23  C42.61C23  C42.64C23  C42.492C23  C42.496C23  C22.64C25  C22.69C25  C22.81C25  C22.82C25  C22.83C25  C22.101C25  C22.102C25  C22.104C25  C22.105C25  C22.110C25  C22.113C25  C22.122C25  C22.123C25  C22.142C25  C22.148C25  C22.149C25  C22.154C25  C22.155C25  C22.156C25
 C2p.2+ 1+4: C42.462C23  C42.466C23  C42.42C23  C42.44C23  C42.53C23  C42.55C23  C42.471C23  C42.475C23 ...
C22.47C24 is a maximal quotient of
C23.227C24  C23.229C24  C23.234C24  C23.235C24  C24.212C23  C24.215C23  C24.217C23  C24.218C23  C23.252C24  C23.255C24  C24.223C23  C24.249C23  C23.316C24  C24.254C23  C23.322C24  C24.269C23  C23.344C24  C23.356C24  C24.278C23  C23.364C24  C24.286C23  C23.367C24  C23.368C24  C24.289C23  C24.293C23  C24.573C23  C23.385C24  C24.300C23  C24.304C23  C23.395C24  C23.397C24  C23.400C24  C23.404C24  C23.407C24  C23.409C24  C23.410C24  C23.412C24  C23.413C24  C24.309C23  C23.416C24  C23.418C24  C23.422C24  C23.425C24  C23.426C24  C23.429C24  C23.430C24  C23.431C24  C4217D4  C23.443C24  C24.327C23  C24.331C23  C24.584C23  C42.36Q8  C23.473C24  C24.340C23  C24.341C23  C23.478C24  C23.479C24  C23.485C24  C24.345C23  C23.490C24  C23.491C24  C23.493C24  C23.494C24  C24.347C23  C23.496C24  C24.348C23  C4222D4  C23.500C24  C4223D4  C23.502C24  C4224D4  C42.38Q8  C23.508C24  C24.395C23  C23.591C24  C24.407C23  C23.603C24  C23.608C24  C24.413C23  C23.618C24  C24.427C23  C23.641C24  C24.432C23  C23.647C24  C23.649C24  C24.435C23  C24.437C23  C23.656C24  C24.438C23  C24.440C23  C24.443C23  C23.666C24  C24.445C23  C23.672C24  C23.676C24  C23.677C24  C23.678C24  C23.679C24  C24.448C23  C23.681C24  C23.682C24  C23.683C24  C23.686C24  C23.687C24  C24.454C23  C23.693C24  C23.695C24  C23.696C24  C23.697C24  C23.700C24  C23.701C24  C23.702C24  C23.703C24
 C42.D2p: C42.172D4  C42.175D4  C42.95D6  C42.104D6  C42.113D6  C42.119D6  C42.153D6  C42.163D6 ...
 C4⋊C4.D2p: C24.268C23  C24.569C23  C23.360C24  C23.390C24  C23.458C24  C23.611C24  C23.625C24  C6.112+ 1+4 ...

Matrix representation of C22.47C24 in GL4(𝔽5) generated by

1000
0100
0040
0004
,
4000
0400
0010
0001
,
0200
3000
0003
0020
,
2000
0200
0002
0020
,
0100
1000
0040
0004
,
1000
0100
0001
0040
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],[0,3,0,0,2,0,0,0,0,0,0,2,0,0,3,0],[2,0,0,0,0,2,0,0,0,0,0,2,0,0,2,0],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0] >;

C22.47C24 in GAP, Magma, Sage, TeX

C_2^2._{47}C_2^4
% in TeX

G:=Group("C2^2.47C2^4");
// GroupNames label

G:=SmallGroup(64,234);
// by ID

G=gap.SmallGroup(64,234);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=1,d^2=b*a=a*b,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=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 C22.47C24 in TeX

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