Copied to
clipboard

G = C22.34C24order 64 = 26

20th central stem extension by C22 of C24

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

Aliases: C23.14C23, C42.39C22, C22.34C24, C2.82+ 1+4, (C4×D4)⋊12C2, C41D46C2, C4⋊D49C2, C42.C25C2, C4.20(C4○D4), C4⋊C4.29C22, (C2×C4).21C23, C42⋊C212C2, (C2×D4).33C22, C22.D46C2, C22⋊C4.3C22, (C22×C4).63C22, C2.17(C2×C4○D4), SmallGroup(64,221)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.34C24
C1C2C22C2×C4C22×C4C42⋊C2 — C22.34C24
C1C22 — C22.34C24
C1C22 — C22.34C24
C1C22 — C22.34C24

Generators and relations for C22.34C24
 G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=b, f2=a, 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: 201 in 120 conjugacy classes, 73 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C2×C4 [×2], C2×C4 [×8], C2×C4 [×6], D4 [×12], C23, C23 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×10], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C41D4, C22.34C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ 1+4 [×2], C22.34C24

Character table of C22.34C24

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M
 size 1111444442222224444444
ρ11111111111111111111111    trivial
ρ21111111-1-1-1-111-1-11-1-1-111-1    linear of order 2
ρ311111-111-1-1-1-1-111-111-11-1-1    linear of order 2
ρ411111-11-1111-1-1-1-1-1-1-111-11    linear of order 2
ρ51111-111-11-1-1-1-111-11-11-11-1    linear of order 2
ρ61111-1111-111-1-1-1-1-1-11-1-111    linear of order 2
ρ71111-1-11-1-111111111-1-1-1-11    linear of order 2
ρ81111-1-1111-1-111-1-11-111-1-1-1    linear of order 2
ρ91111-11-1-1111-1-1-1-1111-11-1-1    linear of order 2
ρ101111-11-11-1-1-1-1-1111-1-111-11    linear of order 2
ρ111111-1-1-1-1-1-1-111-1-1-1111111    linear of order 2
ρ121111-1-1-111111111-1-1-1-111-1    linear of order 2
ρ13111111-111-1-111-1-1-11-1-1-1-11    linear of order 2
ρ14111111-1-1-1111111-1-111-1-1-1    linear of order 2
ρ1511111-1-11-111-1-1-1-111-11-11-1    linear of order 2
ρ1611111-1-1-11-1-1-1-1111-11-1-111    linear of order 2
ρ172-22-2000002i-2i2-22i-2i0000000    complex lifted from C4○D4
ρ182-22-200000-2i2i2-2-2i2i0000000    complex lifted from C4○D4
ρ192-22-200000-2i2i-222i-2i0000000    complex lifted from C4○D4
ρ202-22-2000002i-2i-22-2i2i0000000    complex lifted from C4○D4
ρ2144-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of C22.34C24
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 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 11)(2 16)(3 9)(4 14)(5 32)(6 21)(7 30)(8 23)(10 26)(12 28)(13 25)(15 27)(17 29)(18 22)(19 31)(20 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 23)(2 32)(3 21)(4 30)(5 14)(6 11)(7 16)(8 9)(10 20)(12 18)(13 19)(15 17)(22 26)(24 28)(25 29)(27 31)
(1 15 27 11)(2 16 28 12)(3 13 25 9)(4 14 26 10)(5 22 20 30)(6 23 17 31)(7 24 18 32)(8 21 19 29)

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

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

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

C22.34C24 is a maximal subgroup of
C42.353C23  C42.356C23  C42.360C23  C42.423C23  C22.44C25  C22.83C25  C22.101C25  C22.102C25  C22.113C25  C22.122C25  C22.123C25  C22.146C25  C22.148C25  C22.155C25  C22.156C25
 C2p.2+ 1+4: C42.406C23  C42.407C23  C4.2- 1+4  C42.26C23  C42.27C23  C22.49C25  C22.97C25  C22.106C25 ...
 C8pD4⋊C2: C42.386C23  C42.388C23  C42.391C23 ...
C22.34C24 is a maximal quotient of
C24.192C23  C23.201C24  C4213D4  C4214D4  C42.33Q8  C23.215C24  C24.204C23  C24.254C23  C23.328C24  C23.345C24  C24.278C23  C23.364C24  C24.290C23  C24.293C23  C23.397C24  C23.407C24  C23.413C24  C23.416C24  C23.422C24  C23.426C24  C23.431C24  C4227D4  C23.524C24  C23.530C24  C23.535C24  C23.544C24  C42.39Q8  C23.548C24  C24.375C23  C23.551C24  C23.554C24  C24.377C23  C23.567C24  C23.571C24  C23.573C24  C24.395C23  C24.406C23  C24.407C23  C23.603C24  C23.606C24  C24.411C23  C24.426C23  C23.640C24  C23.641C24  C23.643C24  C24.434C23  C23.649C24  C23.652C24  C24.437C23  C23.656C24  C24.440C23  C23.668C24  C23.673C24  C23.677C24  C24.448C23  C23.686C24  C23.691C24  C23.693C24  C23.697C24  C23.700C24  C23.703C24  C23.728C24  C23.729C24  C23.736C24  C23.737C24
 C42.D2p: C42.188D4  C42.194D4  C42.100D6  C42.116D6  C42.155D6  C42.168D6  C42.100D10  C42.116D10 ...
 C4⋊C4.D2p: C23.354C24  C23.390C24  C23.607C24  C23.611C24  C6.442+ 1+4  C6.472+ 1+4  C6.662+ 1+4  C10.442+ 1+4 ...

Matrix representation of C22.34C24 in GL6(𝔽5)

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

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

C22.34C24 in GAP, Magma, Sage, TeX

C_2^2._{34}C_2^4
% in TeX

G:=Group("C2^2.34C2^4");
// GroupNames label

G:=SmallGroup(64,221);
// by ID

G=gap.SmallGroup(64,221);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=1,d^2=b,f^2=a,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 C22.34C24 in TeX

׿
×
𝔽