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G = C22.53C24order 64 = 26

39th central stem extension by C22 of C24

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

Aliases: C42.54C22, C23.22C23, C22.53C24, C2.202+ 1+4, (C4×D4)⋊22C2, (C4×Q8)⋊17C2, C41D4.7C2, C4.47(C4○D4), C4.4D414C2, C4⋊C4.77C22, (C2×C4).35C23, (C2×D4).37C22, (C2×Q8).67C22, C22.D412C2, C22⋊C4.25C22, (C22×C4).15C22, C2.31(C2×C4○D4), SmallGroup(64,240)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.53C24
C1C2C22C2×C4C42C4×Q8 — C22.53C24
C1C22 — C22.53C24
C1C22 — C22.53C24
C1C22 — C22.53C24

Generators and relations for C22.53C24
 G = < a,b,c,d,e,f | a2=b2=1, c2=e2=b, d2=ba=ab, f2=a, dcd-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, ef=fe >

Subgroups: 181 in 118 conjugacy classes, 75 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C2×C4, C2×C4 [×10], C2×C4 [×4], D4 [×10], Q8 [×4], C23 [×4], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C4×D4 [×4], C4×Q8 [×2], C22.D4 [×4], C4.4D4 [×4], C41D4, C22.53C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×4], C24, C2×C4○D4 [×2], 2+ 1+4, C22.53C24

Character table of C22.53C24

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 1111444422222222222244444
ρ11111111111111111111111111    trivial
ρ21111-1-1-1-1-1-1-111111111-1111-1-1    linear of order 2
ρ31111-111-1-11-1-1-11-1-1-11-111-111-1    linear of order 2
ρ411111-1-111-11-1-11-1-1-11-1-11-11-11    linear of order 2
ρ51111-11-11111-111-1-111-11-11-1-1-1    linear of order 2
ρ611111-11-1-1-1-1-111-1-111-1-1-11-111    linear of order 2
ρ7111111-1-1-11-11-1111-1111-1-1-1-11    linear of order 2
ρ81111-1-1111-111-1111-111-1-1-1-11-1    linear of order 2
ρ911111111-1-1-111-1-111-1-1-11-1-1-1-1    linear of order 2
ρ101111-1-1-1-111111-1-111-1-111-1-111    linear of order 2
ρ111111-111-11-11-1-1-11-1-1-11-111-1-11    linear of order 2
ρ1211111-1-11-11-1-1-1-11-1-1-11111-11-1    linear of order 2
ρ131111-11-11-1-1-1-11-11-11-11-1-1-1111    linear of order 2
ρ1411111-11-1111-11-11-11-111-1-11-1-1    linear of order 2
ρ15111111-1-11-111-1-1-11-1-1-1-1-1111-1    linear of order 2
ρ161111-1-111-11-11-1-1-11-1-1-11-111-11    linear of order 2
ρ172-2-22000000020-2i-2i-202i2i000000    complex lifted from C4○D4
ρ182-2-220000000-202i-2i20-2i2i000000    complex lifted from C4○D4
ρ192-22-20000-2i2i2i02000-200-2i00000    complex lifted from C4○D4
ρ202-22-200002i-2i-2i02000-2002i00000    complex lifted from C4○D4
ρ212-2-220000000202i2i-20-2i-2i000000    complex lifted from C4○D4
ρ222-22-20000-2i-2i2i0-20002002i00000    complex lifted from C4○D4
ρ232-2-220000000-20-2i2i202i-2i000000    complex lifted from C4○D4
ρ242-22-200002i2i-2i0-2000200-2i00000    complex lifted from C4○D4
ρ2544-4-4000000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of C22.53C24
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 29)(6 30)(7 31)(8 32)(13 25)(14 26)(15 27)(16 28)(17 24)(18 21)(19 22)(20 23)
(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 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 28 11 14)(2 13 12 27)(3 26 9 16)(4 15 10 25)(5 23 31 18)(6 17 32 22)(7 21 29 20)(8 19 30 24)
(1 19 3 17)(2 18 4 20)(5 15 7 13)(6 14 8 16)(9 22 11 24)(10 21 12 23)(25 29 27 31)(26 32 28 30)
(1 25 9 13)(2 26 10 14)(3 27 11 15)(4 28 12 16)(5 19 29 22)(6 20 30 23)(7 17 31 24)(8 18 32 21)

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

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

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

C22.53C24 is a maximal subgroup of
C42.181C23  C42.191C23  C42.201C23  C42.502C23  C42.506C23  C42.511C23  C42.512C23  C42.514C23  C42.516C23  C22.69C25  C22.96C25  C22.99C25  C22.102C25  C22.111C25  C22.113C25  C22.122C25  C22.146C25  C22.155C25
 C42.D2p: C42.13D4  C42.114D6  C42.136D6  C42.143D6  C42.166D6  C42.114D10  C42.136D10  C42.143D10 ...
 C2p.2+ 1+4: C42.528C23  C42.530C23  C42.74C23  C42.75C23  C42.531C23  C42.533C23  C22.70C25  C22.95C25 ...
C22.53C24 is a maximal quotient of
C23.237C24  C24.212C23  C24.219C23  C24.223C23  C23.345C24  C24.271C23  C23.348C24  C24.276C23  C23.359C24  C23.411C24  C23.413C24  C23.416C24  C23.417C24  C4221D4  C23.457C24  C42.36Q8  C42.37Q8  C24.339C23  C24.346C23  C23.493C24  C24.347C23  C23.496C24  C24.348C23  C23.500C24  C23.502C24  C24.355C23  C24.411C23  C24.412C23  C23.612C24  C23.624C24  C23.651C24  C23.652C24  C23.654C24  C23.671C24  C23.673C24  C23.694C24  C23.696C24  C23.697C24  C23.698C24  C23.703C24  C24.456C23  C23.707C24  C23.708C24
 C42.D2p: C42.171D4  C42.178D4  C42.180D4  C42.182D4  C42.114D6  C42.136D6  C42.143D6  C42.166D6 ...
 C4⋊C4.D2p: C24.220C23  C24.279C23  C6.672+ 1+4  C10.672+ 1+4  C14.672+ 1+4 ...

Matrix representation of C22.53C24 in GL4(𝔽5) generated by

1000
0100
0040
0004
,
4000
0400
0010
0001
,
2000
3300
0034
0032
,
2000
0200
0030
0032
,
4300
1100
0040
0004
,
4000
0400
0042
0041
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,3,0,0,0,3,0,0,0,0,3,3,0,0,4,2],[2,0,0,0,0,2,0,0,0,0,3,3,0,0,0,2],[4,1,0,0,3,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,4,0,0,2,1] >;

C22.53C24 in GAP, Magma, Sage, TeX

C_2^2._{53}C_2^4
% in TeX

G:=Group("C2^2.53C2^4");
// GroupNames label

G:=SmallGroup(64,240);
// by ID

G=gap.SmallGroup(64,240);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,295,650,158,297,69]);
// Polycyclic

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

Export

Character table of C22.53C24 in TeX

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