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G = C22.49C24order 64 = 26

35th central stem extension by C22 of C24

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

Aliases: C42.50C22, C23.20C23, C22.49C24, C2.182+ 1+4, C4⋊Q816C2, (C4×D4)⋊21C2, C4⋊D416C2, C4.38(C4○D4), C4.4D413C2, C4⋊C4.83C22, (C2×C4).32C23, C42⋊C217C2, (C2×D4).36C22, (C2×Q8).33C22, C22⋊C4.24C22, (C22×C4).74C22, C2.28(C2×C4○D4), SmallGroup(64,236)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.49C24
C1C2C22C2×C4C22×C4C42⋊C2 — C22.49C24
C1C22 — C22.49C24
C1C22 — C22.49C24
C1C22 — C22.49C24

Generators and relations for C22.49C24
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=b, e2=f2=a, ab=ba, 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 [×8], D4 [×8], Q8 [×2], C23 [×4], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C22.49C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×4], C24, C2×C4○D4 [×2], 2+ 1+4, C22.49C24

Character table of C22.49C24

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 1111444422222222222244444
ρ11111111111111111111111111    trivial
ρ21111111-11-11-1-1-1-1-11-1-11-1-11-11    linear of order 2
ρ31111-1-1-1-111-1-1-1-11-1-1-1-1111111    linear of order 2
ρ41111-1-1-111-1-1111-11-1111-1-11-11    linear of order 2
ρ51111-111-111-1-11111-1-1111-1-1-1-1    linear of order 2
ρ61111-11111-1-11-1-1-1-1-11-11-11-11-1    linear of order 2
ρ711111-1-111111-1-11-111-111-1-1-1-1    linear of order 2
ρ811111-1-1-11-11-111-111-111-11-11-1    linear of order 2
ρ9111111-1-1-1-1-11-11-1-1-111-1111-1-1    linear of order 2
ρ10111111-11-11-1-11-111-1-1-1-1-1-111-1    linear of order 2
ρ111111-1-111-1-11-11-1-111-1-1-1111-1-1    linear of order 2
ρ121111-1-11-1-1111-111-1111-1-1-111-1    linear of order 2
ρ131111-11-11-1-11-1-11-1-11-11-11-1-111    linear of order 2
ρ141111-11-1-1-11111-11111-1-1-11-1-11    linear of order 2
ρ1511111-11-1-1-1-111-1-11-11-1-11-1-111    linear of order 2
ρ1611111-111-11-1-1-111-1-1-11-1-11-1-11    linear of order 2
ρ172-2-22000020002i2i0-2i00-2i-200000    complex lifted from C4○D4
ρ182-2-220000-2000-2i2i02i00-2i200000    complex lifted from C4○D4
ρ192-22-2000002-2i-2i00-202i2i0000000    complex lifted from C4○D4
ρ202-2-220000-20002i-2i0-2i002i200000    complex lifted from C4○D4
ρ212-22-200000-22i-2i0020-2i2i0000000    complex lifted from C4○D4
ρ222-22-200000-2-2i2i00202i-2i0000000    complex lifted from C4○D4
ρ232-2-2200002000-2i-2i02i002i-200000    complex lifted from C4○D4
ρ242-22-20000022i2i00-20-2i-2i0000000    complex lifted from C4○D4
ρ2544-4-4000000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of C22.49C24
On 32 points
Generators in S32
(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 24)(10 21)(11 22)(12 23)(25 30)(26 31)(27 32)(28 29)
(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 31)(2 27)(3 29)(4 25)(5 21)(6 11)(7 23)(8 9)(10 18)(12 20)(13 28)(14 30)(15 26)(16 32)(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)
(1 8 15 17)(2 5 16 18)(3 6 13 19)(4 7 14 20)(9 28 24 29)(10 25 21 30)(11 26 22 31)(12 27 23 32)
(1 17 15 8)(2 5 16 18)(3 19 13 6)(4 7 14 20)(9 31 24 26)(10 27 21 32)(11 29 22 28)(12 25 23 30)

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

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

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

C22.49C24 is a maximal subgroup of
C42.185C23  C42.195C23  C42.207C23  C42.489C23  C42.490C23  C42.61C23  C42.62C23  C42.495C23  C42.498C23  C22.64C25  C22.82C25  C22.83C25  C22.107C25  C22.113C25  C22.133C25  C22.134C25  C22.154C25  C22.155C25
 C42.D2p: C42.17D4  C42.97D6  C42.117D6  C42.138D6  C42.179D6  C42.97D10  C42.117D10  C42.138D10 ...
 C2p.2+ 1+4: C42.468C23  C42.470C23  C42.54C23  C42.56C23  C42.474C23  C42.478C23  C22.72C25  C22.95C25 ...
C22.49C24 is a maximal quotient of
C24.208C23  C23.236C24  C24.217C23  C23.251C24  C24.221C23  C24.249C23  C24.254C23  C23.359C24  C23.374C24  C24.293C23  C23.385C24  C23.395C24  C23.397C24  C23.400C24  C23.408C24  C23.413C24  C23.418C24  C23.426C24  C23.429C24  C23.431C24  C23.432C24  C4218D4  C23.455C24  C42.37Q8  C23.472C24  C24.338C23  C23.493C24  C4225D4  C4226D4  C429Q8  C24.393C23  C24.406C23  C23.631C24  C23.633C24  C23.637C24  C24.430C23  C23.656C24  C23.669C24  C23.675C24  C23.681C24  C23.683C24  C24.450C23  C23.685C24  C23.695C24  C23.703C24  C23.708C24  C23.709C24
 C42.D2p: C42.173D4  C42.97D6  C42.117D6  C42.138D6  C42.179D6  C42.97D10  C42.117D10  C42.138D10 ...
 C4⋊C4.D2p: C24.267C23  C23.391C24  C6.452+ 1+4  C10.452+ 1+4  C14.452+ 1+4 ...

Matrix representation of C22.49C24 in GL4(𝔽5) generated by

4000
0400
0010
0001
,
1000
0100
0040
0004
,
4000
0100
0001
0010
,
0400
4000
0030
0003
,
2000
0200
0010
0004
,
3000
0200
0010
0001
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],[4,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,4,0,0,4,0,0,0,0,0,3,0,0,0,0,3],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,4],[3,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1] >;

C22.49C24 in GAP, Magma, Sage, TeX

C_2^2._{49}C_2^4
% in TeX

G:=Group("C2^2.49C2^4");
// GroupNames label

G:=SmallGroup(64,236);
// by ID

G=gap.SmallGroup(64,236);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,86,297,69]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=1,d^2=b,e^2=f^2=a,a*b=b*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 C22.49C24 in TeX

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