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G = C22.33C24order 64 = 26

19th central stem extension by C22 of C24

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

Aliases: C22.33C24, C23.39C23, C42.38C22, C2.72+ 1+4, C2.52- 1+4, (C4×D4)⋊11C2, C22⋊Q88C2, C4⋊D4.8C2, C42.C24C2, C422C22C2, C4⋊C4.28C22, (C2×C4).20C23, (C2×D4).66C22, C22.5(C4○D4), C22.D45C2, (C2×Q8).29C22, C22⋊C4.16C22, (C22×C4).13C22, (C2×C4⋊C4)⋊19C2, C2.16(C2×C4○D4), SmallGroup(64,220)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.33C24
C1C2C22C23C22×C4C2×C4⋊C4 — C22.33C24
C1C22 — C22.33C24
C1C22 — C22.33C24
C1C22 — C22.33C24

Generators and relations for C22.33C24
 G = < a,b,c,d,e,f | a2=b2=c2=f2=1, d2=b, e2=a, ab=ba, dcd-1=fcf=ac=ca, ede-1=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, df=fd, ef=fe >

Subgroups: 161 in 109 conjugacy classes, 73 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C2×C4 [×2], C2×C4 [×10], C2×C4 [×6], D4 [×5], Q8, C23, C23 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×4], C42.C2 [×2], C422C2 [×2], C22.33C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.33C24

Character table of C22.33C24

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 1111224422224444444444
ρ11111111111111111111111    trivial
ρ21111-1-11-111-1-11-1-111-1-111-1    linear of order 2
ρ31111-1-1-1-111-1-11-11-1-1111-11    linear of order 2
ρ4111111-11111111-1-1-1-1-11-1-1    linear of order 2
ρ5111111-1-11111-1-1-11111-1-1-1    linear of order 2
ρ61111-1-1-1111-1-1-11111-1-1-1-11    linear of order 2
ρ71111-1-11111-1-1-11-1-1-111-11-1    linear of order 2
ρ81111111-11111-1-11-1-1-1-1-111    linear of order 2
ρ91111111-1-1-1-1-11111-1-11-1-1-1    linear of order 2
ρ101111-1-111-1-1111-1-11-11-1-1-11    linear of order 2
ρ111111-1-1-11-1-1111-11-11-11-11-1    linear of order 2
ρ12111111-1-1-1-1-1-111-1-111-1-111    linear of order 2
ρ13111111-11-1-1-1-1-1-1-11-1-11111    linear of order 2
ρ141111-1-1-1-1-1-111-1111-11-111-1    linear of order 2
ρ151111-1-11-1-1-111-11-1-11-111-11    linear of order 2
ρ1611111111-1-1-1-1-1-11-111-11-1-1    linear of order 2
ρ172-22-2-22002i-2i-2i2i0000000000    complex lifted from C4○D4
ρ182-22-2-2200-2i2i2i-2i0000000000    complex lifted from C4○D4
ρ192-22-22-200-2i2i-2i2i0000000000    complex lifted from C4○D4
ρ202-22-22-2002i-2i2i-2i0000000000    complex lifted from C4○D4
ρ2144-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.33C24
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(2 28)(4 26)(5 32)(6 8)(7 30)(9 15)(11 13)(17 21)(18 20)(19 23)(22 24)(29 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)
(1 31 27 6)(2 7 28 32)(3 29 25 8)(4 5 26 30)(9 17 15 23)(10 24 16 18)(11 19 13 21)(12 22 14 20)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)

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

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

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

C22.33C24 is a maximal subgroup of
C4⋊C4.D4  C4⋊C4.6D4  C4⋊C4.12D4  C4⋊C4.18D4  C22.44C25  C22.48C25  C22.80C25  C22.81C25  C22.94C25  C22.102C25  C23.144C24  C22.110C25  C22.122C25  C22.123C25  C22.124C25  C22.125C25  C22.127C25  C22.128C25  C22.130C25  C22.142C25  C22.149C25  C22.151C25  C22.153C25  C22.155C25  C22.156C25
 C2p.2- 1+4: C22.50C25  C22.96C25  C22.104C25  C22.141C25  C22.143C25  C22.146C25  C22.148C25  C22.152C25 ...
C22.33C24 is a maximal quotient of
C23.195C24  C24.547C23  C24.195C23  C24.198C23  C23.211C24  C23.214C24  C24.203C23  C24.204C23  C23.218C24  C24.563C23  C24.258C23  C24.269C23  C23.349C24  C24.276C23  C24.283C23  C24.286C23  C24.573C23  C23.388C24  C24.577C23  C23.417C24  C24.313C23  C24.326C23  C24.332C23  C23.473C24  C24.339C23  C24.341C23  C23.485C24  C23.488C24  C23.490C24  C23.494C24  C23.496C24  C24.587C23  C24.589C23  C23.527C24  C23.530C24  C4230D4  C24.374C23  C24.592C23  C23.543C24  C23.546C24  C24.375C23  C23.550C24  C23.551C24  C24.376C23  C23.553C24  C23.554C24  C23.555C24  C24.378C23  C4211Q8  C23.567C24  C23.580C24  C23.581C24  C24.394C23  C23.589C24  C23.591C24  C23.593C24  C23.595C24  C24.403C23  C23.605C24  C23.606C24  C23.608C24  C23.618C24  C24.418C23  C23.624C24  C23.627C24  C24.420C23  C23.632C24  C23.637C24  C24.426C23  C24.427C23  C23.640C24  C23.643C24  C24.430C23  C23.645C24  C24.432C23  C23.647C24  C24.435C23  C24.438C23  C24.443C23  C23.667C24  C23.669C24  C24.445C23  C23.671C24  C23.673C24  C23.675C24  C23.676C24  C23.677C24  C23.681C24  C23.682C24  C23.683C24  C23.685C24  C23.686C24  C23.687C24  C23.689C24  C24.454C23  C23.691C24  C23.692C24  C23.693C24  C23.694C24  C23.695C24  C23.696C24  C23.698C24  C23.702C24  C23.705C24  C23.707C24  C23.708C24  C23.709C24  C23.710C24  C23.724C24  C23.726C24  C23.727C24  C23.734C24  C23.735C24  C23.736C24  C23.737C24  C23.738C24  C23.739C24
 C42.D2p: C42.190D4  C42.198D4  C42.118D6  C42.150D6  C42.161D6  C42.118D10  C42.150D10  C42.161D10 ...
 C4⋊C4.D2p: C23.360C24  C24.572C23  C23.456C24  C23.458C24  C23.590C24  C24.401C23  C23.607C24  C23.611C24 ...

Matrix representation of C22.33C24 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
040000
001000
000400
000010
000004
,
200000
020000
000004
000040
000400
004000
,
010000
100000
000010
000001
004000
000400
,
100000
010000
000100
001000
000001
000010

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C22.33C24 in GAP, Magma, Sage, TeX

C_2^2._{33}C_2^4
% in TeX

G:=Group("C2^2.33C2^4");
// GroupNames label

G:=SmallGroup(64,220);
// by ID

G=gap.SmallGroup(64,220);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,188,86,579]);
// Polycyclic

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

Export

Character table of C22.33C24 in TeX

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