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G = C22.35C24order 64 = 26

21st central stem extension by C22 of C24

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

Aliases: C22.35C24, C23.15C23, C42.40C22, C2.62- 1+4, C4⋊Q810C2, (C4×Q8)⋊9C2, C422C2.C2, C42.C26C2, C22⋊Q8.8C2, C4.21(C4○D4), C4⋊C4.30C22, (C2×C4).22C23, (C2×Q8).60C22, C42⋊C2.12C2, C22⋊C4.17C22, (C22×C4).64C22, C2.18(C2×C4○D4), SmallGroup(64,222)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.35C24
C1C2C22C2×C4C22×C4C42⋊C2 — C22.35C24
C1C22 — C22.35C24
C1C22 — C22.35C24
C1C22 — C22.35C24

Generators and relations for C22.35C24
 G = < a,b,c,d,e,f | a2=b2=e2=1, c2=f2=a, d2=b, ab=ba, dcd-1=fcf-1=ac=ca, ede=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, df=fd, ef=fe >

Subgroups: 121 in 96 conjugacy classes, 73 normal (13 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×13], C22, C22 [×3], C2×C4 [×2], C2×C4 [×12], C2×C4 [×2], Q8 [×4], C23, C42 [×2], C42 [×4], C22⋊C4 [×6], C4⋊C4 [×20], C22×C4, C2×Q8 [×2], C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2, C42.C2 [×4], C422C2 [×4], C4⋊Q8, C22.35C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2- 1+4 [×2], C22.35C24

Character table of C22.35C24

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 1111422222244444444444
ρ11111111111111111111111    trivial
ρ21111-1111111-111-1-1-1-1-111-1    linear of order 2
ρ311111111111-1-1-1-1-1111-1-1-1    linear of order 2
ρ41111-11111111-1-111-1-1-1-1-11    linear of order 2
ρ511111-1-1-11-11-111111-1-1-1-1-1    linear of order 2
ρ61111-1-1-1-11-11111-1-1-111-1-11    linear of order 2
ρ711111-1-1-11-111-1-1-1-11-1-1111    linear of order 2
ρ81111-1-1-1-11-11-1-1-111-11111-1    linear of order 2
ρ91111-11-11-1-1-1-11-1-1111-1-111    linear of order 2
ρ10111111-11-1-1-111-11-1-1-11-11-1    linear of order 2
ρ111111-11-11-1-1-11-111-111-11-1-1    linear of order 2
ρ12111111-11-1-1-1-1-11-11-1-111-11    linear of order 2
ρ131111-1-11-1-11-111-1-111-111-1-1    linear of order 2
ρ1411111-11-1-11-1-11-11-1-11-11-11    linear of order 2
ρ151111-1-11-1-11-1-1-111-11-11-111    linear of order 2
ρ1611111-11-1-11-11-11-11-11-1-11-1    linear of order 2
ρ172-22-20-2i2i2i-2-2i200000000000    complex lifted from C4○D4
ρ182-22-20-2i-2i2i22i-200000000000    complex lifted from C4○D4
ρ192-22-202i-2i-2i-22i200000000000    complex lifted from C4○D4
ρ202-22-202i2i-2i2-2i-200000000000    complex lifted from C4○D4
ρ2144-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ224-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.35C24
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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)
(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 11)(2 30 28 10)(3 29 25 9)(4 32 26 12)(5 21 20 13)(6 24 17 16)(7 23 18 15)(8 22 19 14)
(2 28)(4 26)(5 7)(6 19)(8 17)(9 11)(10 32)(12 30)(14 22)(16 24)(18 20)(29 31)
(1 21 3 23)(2 24 4 22)(5 9 7 11)(6 12 8 10)(13 25 15 27)(14 28 16 26)(17 32 19 30)(18 31 20 29)

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

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

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

C22.35C24 is a maximal subgroup of
C42.354C23  C42.357C23  C42.361C23  C42.386C23  C42.389C23  C42.390C23  C42.408C23  C42.409C23  C22.44C25  C22.84C25  C22.101C25  C22.105C25  C22.110C25  C22.127C25  C22.128C25  C22.130C25  C22.142C25  C22.151C25  C22.153C25  C22.155C25
 C2p.2- 1+4: C42.424C23  C42.25C23  C42.27C23  C42.28C23  C22.50C25  C22.98C25  C22.104C25  C23.146C24 ...
C22.35C24 is a maximal quotient of
C24.192C23  C23.202C24  C42.33Q8  C24.203C23  C23.218C24  C23.321C24  C23.346C24  C24.295C23  C23.396C24  C23.414C24  C23.424C24  C23.428C24  C23.433C24  C23.525C24  C23.545C24  C42.39Q8  C23.550C24  C24.376C23  C23.554C24  C23.555C24  C4211Q8  C23.589C24  C24.405C23  C23.622C24  C24.427C23  C24.430C23  C23.647C24  C23.654C24  C23.655C24  C23.658C24  C23.662C24  C23.666C24  C23.669C24  C23.672C24  C23.676C24  C23.683C24  C23.689C24  C23.694C24  C23.698C24  C23.702C24  C23.731C24  C23.733C24  C23.736C24  C23.739C24
 C42.D2p: C42.159D4  C42.161D4  C42.186D4  C42.188D4  C42.191D4  C42.192D4  C42.195D4  C42.198D4 ...
 C4⋊C4.D2p: C23.329C24  C23.353C24  C23.362C24  C23.369C24  C23.375C24  C24.301C23  C23.406C24  C23.419C24 ...

Matrix representation of C22.35C24 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
100000
000010
000001
004000
000400
,
300000
030000
004300
001100
000012
000044
,
100000
040000
001000
004400
000040
000011
,
400000
040000
002000
000200
000030
000003

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C22.35C24 in GAP, Magma, Sage, TeX

C_2^2._{35}C_2^4
% in TeX

G:=Group("C2^2.35C2^4");
// GroupNames label

G:=SmallGroup(64,222);
// by ID

G=gap.SmallGroup(64,222);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,199,650,188,579,69]);
// Polycyclic

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

Export

Character table of C22.35C24 in TeX

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