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G = C22.19C24order 64 = 26

5th central stem extension by C22 of C24

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

Aliases: C425C22, C23.6C23, C22.19C24, C24.32C22, C4C22≀C2, (C4×D4)⋊7C2, (C2×C4)⋊11D4, C4(C4⋊D4), (C23×C4)⋊6C2, C4.81(C2×D4), C4(C22⋊Q8), C22≀C29C2, C4⋊D419C2, C4⋊C412C22, C22⋊Q821C2, (C2×Q8)⋊9C22, C2.8(C22×D4), C42⋊C28C2, (C2×D4)⋊11C22, C221(C4○D4), (C2×C4).13C23, (C22×C4)⋊6C22, C22.19(C2×D4), C22⋊C415C22, C4(C22.D4), C22.D415C2, (C2×C4)C22≀C2, (C2×C4○D4)⋊2C2, C2.8(C2×C4○D4), (C2×C4)(C22⋊Q8), SmallGroup(64,206)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.19C24
C1C2C22C2×C4C22×C4C23×C4 — C22.19C24
C1C22 — C22.19C24
C1C2×C4 — C22.19C24
C1C22 — C22.19C24

Generators and relations for C22.19C24
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f2=a, ab=ba, dcd=ac=ca, ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 249 in 165 conjugacy classes, 85 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C22.19C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24

Character table of C22.19C24

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111222222441111222222444444
ρ11111111111111111111111111111    trivial
ρ211111111111-1-1-1-1-1-1-1-1-1-1-1-11-11-11    linear of order 2
ρ31111111111-1-11111111111-1-1-1-1-1-1    linear of order 2
ρ41111111111-11-1-1-1-1-1-1-1-1-1-11-11-11-1    linear of order 2
ρ51111-11-11-1-1-1-1-1-1-1-11-111-11-11111-1    linear of order 2
ρ61111-11-11-1-1-111111-11-1-11-111-11-1-1    linear of order 2
ρ71111-11-11-1-111-1-1-1-11-111-111-1-1-1-11    linear of order 2
ρ81111-11-11-1-11-11111-11-1-11-1-1-11-111    linear of order 2
ρ911111-1-1-1-111-111111-1-1-1-11111-1-1-1    linear of order 2
ρ1011111-1-1-1-1111-1-1-1-1-11111-1-11-1-11-1    linear of order 2
ρ1111111-1-1-1-11-1111111-1-1-1-11-1-1-1111    linear of order 2
ρ1211111-1-1-1-11-1-1-1-1-1-1-11111-11-111-11    linear of order 2
ρ131111-1-11-11-1-11-1-1-1-111-1-111-111-1-11    linear of order 2
ρ141111-1-11-11-1-1-11111-1-111-1-111-1-111    linear of order 2
ρ151111-1-11-11-11-1-1-1-1-111-1-1111-1-111-1    linear of order 2
ρ161111-1-11-11-1111111-1-111-1-1-1-111-1-1    linear of order 2
ρ1722-2-200-2020002-22-200-2200000000    orthogonal lifted from D4
ρ1822-2-20020-2000-22-2200-2200000000    orthogonal lifted from D4
ρ1922-2-200-202000-22-22002-200000000    orthogonal lifted from D4
ρ2022-2-20020-20002-22-2002-200000000    orthogonal lifted from D4
ρ212-22-20-20200002i2i-2i-2i02i00-2i0000000    complex lifted from C4○D4
ρ222-2-2220000-2002i-2i-2i2i-2i00002i000000    complex lifted from C4○D4
ρ232-22-2020-200002i2i-2i-2i0-2i002i0000000    complex lifted from C4○D4
ρ242-2-2220000-200-2i2i2i-2i2i0000-2i000000    complex lifted from C4○D4
ρ252-2-22-20000200-2i2i2i-2i-2i00002i000000    complex lifted from C4○D4
ρ262-2-22-200002002i-2i-2i2i2i0000-2i000000    complex lifted from C4○D4
ρ272-22-20-2020000-2i-2i2i2i0-2i002i0000000    complex lifted from C4○D4
ρ282-22-2020-20000-2i-2i2i2i02i00-2i0000000    complex lifted from C4○D4

Permutation representations of C22.19C24
On 16 points - transitive group 16T117
Generators in S16
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 11)(10 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,3),(2,4),(5,15),(6,16),(7,13),(8,14),(9,11),(10,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,117);

C22.19C24 is a maximal subgroup of
C4○C2≀C4  M4(2)⋊22D4  C42.297C23  C42.298C23  C42.299C23  C22.33C25  C22.38C25  C22.44C25  C22.48C25  C22.49C25  D4×C4○D4  C22.75C25  C22.76C25  C22.77C25  C22.78C25  C22.79C25  C22.81C25  C22.82C25  C22.83C25  C22.84C25  C22.94C25  C22.95C25  C22.118C25  C22.122C25  C22.123C25  C22.124C25  C22.125C25  C22.126C25  C22.127C25  C22.128C25  C22.129C25  C22.130C25  C22.131C25
 C24.D2p: C24.53D4  C24.150D4  C24.58D4  C24.59D4  C24.72D4  (C2×C4)≀C2  C427D4  C422D4 ...
 (C2×C4p)⋊D4: (C2×C8)⋊D4  C42.264C23  C42.265C23  C4210D6  (C2×D4)⋊43D6  C428D10  (C2×C20)⋊15D4  C428D14 ...
 (C2×D4)⋊D2p: C23.7C24  C22.73C25  C22.74C25  C4⋊C421D6  C4⋊C421D10  C4⋊C421D14 ...
 D2p⋊(C4○D4): C22.64C25  C22.70C25  C22.102C25  C22.108C25  C23.144C24  C4214D6  C4⋊C426D6  C4⋊C428D6 ...
C22.19C24 is a maximal quotient of
C25.85C22  C23.178C24  C23.179C24  C4×C22≀C2  C4×C22.D4  C4×C22⋊Q8  C23.288C24  C23.295C24  C42.162D4  C42.163D4  C425Q8  C24.243C23  C23.311C24  C23.313C24  C23.318C24  C24.276C23  C24.278C23  C24.279C23  C23.359C24  C23.360C24  C23.362C24  C23.364C24  C24.285C23  C24.286C23  C23.368C24  C23.369C24  C24.289C23  C23.372C24  C24.572C23  C23.374C24  C23.375C24  C24.293C23  C24.295C23  C23.380C24  C23.382C24  C24.576C23  C23.385C24  C23.434C24  C42.165D4  C23.439C24  C23.461C24  C42.172D4  C23.479C24  C42.178D4  C42.179D4  C23.483C24  C42.181D4  C23.491C24  C42.182D4  C42.183D4  C23.500C24  C23.502C24  C42.184D4  C42.185D4  C23.530C24  C42.189D4  C42.190D4  C42.191D4  C23.535C24  C42.192D4  C24.374C23  C2413D4  C248Q8  C42.439D4  C23.753C24  C24.598C23  C24.599C23  C42.221D4  C42.222D4  C42.384D4  C42.223D4  C42.224D4  C42.225D4  C42.450D4  C42.451D4  C42.226D4  C42.227D4  C42.228D4  C42.229D4  C42.230D4  C42.231D4  C42.232D4  C42.233D4  C42.234D4  C42.235D4  C42.352C23  C42.353C23  C42.354C23  C42.355C23  C42.356C23  C42.357C23  C42.358C23  C42.359C23  C42.360C23  C42.361C23
 C24.D2p: C24.166D4  C24.67D6  C24.83D6  C24.56D10  C24.72D10  C24.56D14  C24.72D14 ...
 C42⋊D2p: C4242D4  C4×C4⋊D4  C4215D4  C4216D4  C4217D4  C4219D4  C4222D4  C4223D4 ...
 C4⋊C4⋊D2p: C24.249C23  C23.315C24  C24.269C23  C23.349C24  C23.356C24  C24.282C23  C24.283C23  C23.367C24 ...
 (C2×D4)⋊D2p: C23.304C24  C24.244C23  C23.324C24  (C2×D4)⋊43D6  (C2×C20)⋊15D4  (C2×C28)⋊15D4 ...

Matrix representation of C22.19C24 in GL4(𝔽5) generated by

1000
0100
0040
0004
,
4000
0400
0040
0004
,
0100
1000
0001
0010
,
4000
0400
0040
0001
,
1000
0400
0040
0001
,
4000
0400
0030
0003
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,3,0,0,0,0,3] >;

C22.19C24 in GAP, Magma, Sage, TeX

C_2^2._{19}C_2^4
% in TeX

G:=Group("C2^2.19C2^4");
// GroupNames label

G:=SmallGroup(64,206);
// by ID

G=gap.SmallGroup(64,206);
# by ID

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

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

Export

Character table of C22.19C24 in TeX

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