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G = C22.46C24order 64 = 26

32nd central stem extension by C22 of C24

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

Aliases: C22.46C24, C23.47C23, C42.47C22, C2.122- 1+4, (C4×Q8)⋊14C2, (C4×D4).11C2, C22⋊Q816C2, C42.C28C2, C422C25C2, C4.35(C4○D4), C4⋊C4.35C22, (C2×C4).56C23, C42⋊C215C2, (C2×D4).70C22, (C2×Q8).64C22, C22.10(C4○D4), C22⋊C4.30C22, (C22×C4).72C22, C22.D4.2C2, (C2×C4⋊C4)⋊21C2, C2.25(C2×C4○D4), SmallGroup(64,233)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.46C24
C1C2C22C23C22×C4C42⋊C2 — C22.46C24
C1C22 — C22.46C24
C1C22 — C22.46C24
C1C22 — C22.46C24

Generators and relations for C22.46C24
 G = < a,b,c,d,e,f | a2=b2=f2=1, c2=e2=a, d2=b, ab=ba, dcd-1=ac=ca, fdf=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 107 conjugacy classes, 75 normal (31 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C2×C4 [×5], C2×C4 [×8], C2×C4 [×8], D4 [×2], Q8 [×2], C23 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×Q8, C2×C4⋊C4, C42⋊C2, C42⋊C2 [×2], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2, C42.C2 [×2], C422C2 [×2], C22.46C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×4], C24, C2×C4○D4 [×2], 2- 1+4, C22.46C24

Character table of C22.46C24

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 1111224222222222244444444
ρ11111111111111111111111111    trivial
ρ21111-1-11-1-11-1-11-11-11-11-1-11-111    linear of order 2
ρ31111111-1-1-1-1-11-1-1-111-1-1111-1-1    linear of order 2
ρ41111-1-1111-11111-111-1-11-11-1-1-1    linear of order 2
ρ51111-1-1-11-11111-1111-1-1-11-111-1    linear of order 2
ρ6111111-1-111-1-1111-111-11-1-1-11-1    linear of order 2
ρ71111-1-1-1-11-1-1-111-1-11-1111-11-11    linear of order 2
ρ8111111-11-1-1111-1-11111-1-1-1-1-11    linear of order 2
ρ91111-1-1-1-11-11-1-11-11-111-1-1111-1    linear of order 2
ρ10111111-11-1-1-11-1-1-1-1-1-11111-11-1    linear of order 2
ρ111111-1-1-11-11-11-1-11-1-11-11-111-11    linear of order 2
ρ12111111-1-1111-1-1111-1-1-1-111-1-11    linear of order 2
ρ131111111-1-1-11-1-1-1-11-1-1-11-1-1111    linear of order 2
ρ141111-1-1111-1-11-11-1-1-11-1-11-1-111    linear of order 2
ρ151111111111-11-111-1-1-11-1-1-11-1-1    linear of order 2
ρ161111-1-11-1-111-1-1-111-11111-1-1-1-1    linear of order 2
ρ172-22-22-200-2i-2i0002i2i0000000000    complex lifted from C4○D4
ρ182-22-2-2200-2i2i0002i-2i0000000000    complex lifted from C4○D4
ρ1922-2-20002002i-2-2i00-2i2i00000000    complex lifted from C4○D4
ρ2022-2-2000-200-2i2-2i002i2i00000000    complex lifted from C4○D4
ρ2122-2-2000200-2i-22i002i-2i00000000    complex lifted from C4○D4
ρ2222-2-2000-2002i22i00-2i-2i00000000    complex lifted from C4○D4
ρ232-22-2-22002i-2i000-2i2i0000000000    complex lifted from C4○D4
ρ242-22-22-2002i2i000-2i-2i0000000000    complex lifted from C4○D4
ρ254-4-44000000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.46C24
On 32 points
Generators in S32
(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)
(1 30)(2 31)(3 32)(4 29)(5 20)(6 17)(7 18)(8 19)(9 21)(10 22)(11 23)(12 24)(13 27)(14 28)(15 25)(16 26)
(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 30 10)(2 21 31 9)(3 24 32 12)(4 23 29 11)(5 28 20 14)(6 27 17 13)(7 26 18 16)(8 25 19 15)
(1 5 3 7)(2 17 4 19)(6 29 8 31)(9 27 11 25)(10 14 12 16)(13 23 15 21)(18 30 20 32)(22 28 24 26)
(1 13)(2 14)(3 15)(4 16)(5 23)(6 24)(7 21)(8 22)(9 18)(10 19)(11 20)(12 17)(25 32)(26 29)(27 30)(28 31)

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

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

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

C22.46C24 is a maximal subgroup of
C42.461C23  C42.465C23  C42.45C23  C42.47C23  C42.50C23  C42.52C23  C42.472C23  C42.476C23  C22.64C25  C22.80C25  C22.81C25  C22.82C25  C22.84C25  C22.94C25  C22.101C25  C22.102C25  C22.105C25  C23.144C24  C22.110C25  C22.113C25  C22.122C25  C22.124C25  C22.127C25  C22.128C25  C22.130C25  C22.131C25  C22.140C25  C22.142C25  C22.153C25  C22.155C25  C22.156C25
 C2p.2- 1+4: C42.485C23  C42.486C23  C42.57C23  C42.58C23  C42.62C23  C42.63C23  C42.492C23  C42.493C23 ...
C22.46C24 is a maximal quotient of
C23.225C24  C23.226C24  C24.208C23  C23.234C24  C23.238C24  C23.241C24  C23.252C24  C23.253C24  C23.255C24  C23.313C24  C23.315C24  C24.252C23  C24.563C23  C23.321C24  C23.323C24  C24.567C23  C23.368C24  C24.289C23  C23.377C24  C24.295C23  C23.379C24  C24.573C23  C24.576C23  C23.385C24  C24.299C23  C23.388C24  C24.577C23  C24.304C23  C23.395C24  C23.396C24  C23.398C24  C24.308C23  C24.579C23  C23.408C24  C23.411C24  C23.414C24  C24.309C23  C23.417C24  C23.420C24  C23.424C24  C23.425C24  C24.315C23  C23.428C24  C23.429C24  C23.430C24  C23.432C24  C23.433C24  C24.326C23  C42.36Q8  C23.473C24  C24.339C23  C24.341C23  C23.485C24  C23.490C24  C23.496C24  C4223D4  C42.38Q8  C4225D4  C429Q8  C24.394C23  C23.595C24  C24.405C23  C23.602C24  C24.426C23  C24.427C23  C23.641C24  C23.643C24  C24.430C23  C23.645C24  C23.647C24  C24.435C23  C23.651C24  C23.654C24  C23.658C24  C24.440C23  C23.662C24  C23.664C24  C24.443C23  C23.666C24  C23.667C24  C23.668C24  C23.669C24  C24.445C23  C23.671C24  C23.672C24  C23.673C24  C23.674C24  C23.675C24  C23.676C24  C23.677C24  C23.679C24  C23.681C24  C23.687C24  C23.691C24  C23.693C24  C23.694C24
 C42.D2p: C42.165D4  C42.172D4  C42.174D4  C42.183D4  C42.184D4  C42.185D4  C42.94D6  C42.96D6 ...
 C4⋊C4.D2p: C24.558C23  C23.244C24  C24.268C23  C24.569C23  C24.279C23  C23.360C24  C23.362C24  C24.572C23 ...

Matrix representation of C22.46C24 in GL4(𝔽5) generated by

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

C22.46C24 in GAP, Magma, Sage, TeX

C_2^2._{46}C_2^4
% in TeX

G:=Group("C2^2.46C2^4");
// GroupNames label

G:=SmallGroup(64,233);
// by ID

G=gap.SmallGroup(64,233);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,199,650,86,297]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=f^2=1,c^2=e^2=a,d^2=b,a*b=b*a,d*c*d^-1=a*c=c*a,f*d*f=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 C22.46C24 in TeX

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