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G = C22.36C24order 64 = 26

22nd central stem extension by C22 of C24

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

Aliases: C23.16C23, C42.41C22, C22.36C24, C2.72- 1+4, C2.92+ 1+4, C4⋊Q811C2, (C4×D4)⋊13C2, (C4×Q8)⋊10C2, C22⋊Q89C2, C4.4D49C2, C4⋊D4.9C2, C422C23C2, C4.22(C4○D4), C4⋊C4.31C22, (C2×C4).23C23, C42⋊C213C2, (C2×D4).67C22, C22.D47C2, C22⋊C4.4C22, (C2×Q8).30C22, (C22×C4).65C22, C2.19(C2×C4○D4), SmallGroup(64,223)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.36C24
C1C2C22C2×C4C22×C4C42⋊C2 — C22.36C24
C1C22 — C22.36C24
C1C22 — C22.36C24
C1C22 — C22.36C24

Generators and relations for C22.36C24
 G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=ba=ab, f2=a, 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: 161 in 108 conjugacy classes, 73 normal (31 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C2×C4 [×6], C2×C4 [×6], C2×C4 [×4], D4 [×4], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4, C4.4D4 [×2], C422C2 [×2], C4⋊Q8, C22.36C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.36C24

Character table of C22.36C24

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 1111444222222444444444
ρ11111111111111111111111    trivial
ρ21111-1-1-1111111-111-1-1-111-1    linear of order 2
ρ31111-1-11-1-11-11-111-111-11-1-1    linear of order 2
ρ4111111-1-1-11-11-1-11-1-1-111-11    linear of order 2
ρ5111111-11-1-1-1-111-111-1-11-1-1    linear of order 2
ρ61111-1-111-1-1-1-11-1-11-1111-11    linear of order 2
ρ71111-1-1-1-11-11-1-11-1-11-11111    linear of order 2
ρ81111111-11-11-1-1-1-1-1-11-111-1    linear of order 2
ρ911111-1-1-11-11-1-1111-11-1-1-11    linear of order 2
ρ101111-111-11-11-1-1-1111-11-1-1-1    linear of order 2
ρ111111-11-11-1-1-1-1111-1-111-11-1    linear of order 2
ρ1211111-111-1-1-1-11-11-11-1-1-111    linear of order 2
ρ1311111-11-1-11-11-11-11-1-11-11-1    linear of order 2
ρ141111-11-1-1-11-11-1-1-1111-1-111    linear of order 2
ρ151111-1111111111-1-1-1-1-1-1-11    linear of order 2
ρ1611111-1-1111111-1-1-1111-1-1-1    linear of order 2
ρ172-22-2000-2i-2-2i22i2i000000000    complex lifted from C4○D4
ρ182-22-2000-2i22i-2-2i2i000000000    complex lifted from C4○D4
ρ192-22-20002i-22i2-2i-2i000000000    complex lifted from C4○D4
ρ202-22-20002i2-2i-22i-2i000000000    complex lifted from C4○D4
ρ214-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2244-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.36C24
On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 21)(13 17)(14 18)(15 19)(16 20)(25 30)(26 31)(27 32)(28 29)
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 19)(14 20)(15 17)(16 18)(25 32)(26 29)(27 30)(28 31)
(1 26)(2 32)(3 28)(4 30)(5 31)(6 27)(7 29)(8 25)(9 15)(10 20)(11 13)(12 18)(14 21)(16 23)(17 24)(19 22)
(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)
(2 6)(4 8)(9 24)(10 12)(11 22)(14 18)(16 20)(21 23)(25 27)(26 29)(28 31)(30 32)
(1 13 5 17)(2 14 6 18)(3 15 7 19)(4 16 8 20)(9 28 22 29)(10 25 23 30)(11 26 24 31)(12 27 21 32)

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

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

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

C22.36C24 is a maximal subgroup of
C42.352C23  C42.355C23  C42.358C23  C42.359C23  C42.385C23  C42.387C23  C42.390C23  C42.391C23  C42.410C23  C42.411C23  C22.44C25  C22.49C25  C22.83C25  C22.84C25  C22.99C25  C22.102C25  C22.103C25  C22.105C25  C23.144C24  C22.110C25  C22.113C25  C22.122C25  C22.124C25  C22.125C25  C22.129C25  C22.130C25  C22.131C25  C22.134C25  C22.135C25  C22.147C25  C22.149C25  C22.150C25  C22.153C25  C22.155C25  C22.157C25
 C2p.2- 1+4: C42.425C23  C42.426C23  C4.2- 1+4  C42.25C23  C42.29C23  C42.30C23  C22.50C25  C22.100C25 ...
C22.36C24 is a maximal quotient of
C24.192C23  C23.201C24  C23.202C24  C4213D4  C424Q8  C23.214C24  C24.203C23  C24.204C23  C24.205C23  C23.322C24  C23.323C24  C23.327C24  C24.271C23  C23.348C24  C23.352C24  C24.282C23  C23.368C24  C24.289C23  C23.374C24  C23.377C24  C23.379C24  C24.304C23  C23.395C24  C23.408C24  C23.409C24  C23.411C24  C23.412C24  C24.309C23  C23.420C24  C24.311C23  C23.425C24  C24.315C23  C23.429C24  C23.430C24  C23.432C24  C4228D4  C23.524C24  C23.525C24  C23.530C24  C24.374C23  C23.544C24  C23.545C24  C42.39Q8  C24.375C23  C23.550C24  C23.551C24  C24.376C23  C23.553C24  C23.554C24  C23.555C24  C4232D4  C24.378C23  C24.379C23  C4211Q8  C23.572C24  C24.393C23  C24.394C23  C23.591C24  C23.592C24  C24.405C23  C23.600C24  C24.407C23  C23.602C24  C23.605C24  C24.412C23  C23.612C24  C23.615C24  C23.617C24  C23.618C24  C24.418C23  C24.421C23  C23.630C24  C23.631C24  C23.637C24  C24.426C23  C24.427C23  C23.641C24  C24.428C23  C23.645C24  C24.432C23  C23.647C24  C23.651C24  C23.654C24  C23.655C24  C24.438C23  C23.658C24  C23.659C24  C24.440C23  C23.662C24  C23.663C24  C23.664C24  C24.443C23  C23.666C24  C23.667C24  C24.445C23  C23.672C24  C23.673C24  C23.674C24  C23.675C24  C23.677C24  C23.679C24  C24.454C23  C23.693C24  C23.695C24  C23.696C24  C23.698C24  C23.699C24  C23.700C24  C23.701C24  C23.729C24  C23.730C24  C23.731C24  C23.732C24  C23.737C24  C23.738C24  C23.739C24
 C42.D2p: C42.159D4  C42.160D4  C42.187D4  C42.189D4  C42.192D4  C42.193D4  C42.99D6  C42.115D6 ...
 C4⋊C4.D2p: C24.259C23  C23.351C24  C24.279C23  C24.285C23  C23.391C24  C23.392C24  C23.574C24  C23.616C24 ...

Matrix representation of C22.36C24 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
430000
010000
000100
001000
004142
004101
,
300000
030000
000010
001413
004000
004101
,
100000
440000
001000
000100
000040
001404
,
100000
010000
000100
004000
001413
001014

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

C22.36C24 in GAP, Magma, Sage, TeX

C_2^2._{36}C_2^4
% in TeX

G:=Group("C2^2.36C2^4");
// GroupNames label

G:=SmallGroup(64,223);
// by ID

G=gap.SmallGroup(64,223);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,188,579,69]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=1,d^2=b*a=a*b,f^2=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 C22.36C24 in TeX

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