p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊5C22, C23.6C23, C22.19C24, C24.32C22, C4○C22≀C2, (C4×D4)⋊7C2, (C2×C4)⋊11D4, C4○(C4⋊D4), (C23×C4)⋊6C2, C4.81(C2×D4), C4○(C22⋊Q8), C22≀C2⋊9C2, C4⋊D4⋊19C2, C4⋊C4⋊12C22, C22⋊Q8⋊21C2, (C2×Q8)⋊9C22, C2.8(C22×D4), C42⋊C2⋊8C2, (C2×D4)⋊11C22, C22⋊1(C4○D4), (C2×C4).13C23, (C22×C4)⋊6C22, C22.19(C2×D4), C22⋊C4⋊15C22, C4○(C22.D4), C22.D4⋊15C2, (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
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, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C22.19C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C22.19C24
Character table of C22.19C24
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | -2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 16 points - transitive group 16T117
(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 C42⋊7D4 C42⋊2D4 ...
(C2×C4p)⋊D4: (C2×C8)⋊D4 C42.264C23 C42.265C23 C42⋊10D6 (C2×D4)⋊43D6 C42⋊8D10 (C2×C20)⋊15D4 C42⋊8D14 ...
(C2×D4)⋊D2p: C23.7C24 C22.73C25 C22.74C25 C4⋊C4⋊21D6 C4⋊C4⋊21D10 C4⋊C4⋊21D14 ...
D2p⋊(C4○D4): C22.64C25 C22.70C25 C22.102C25 C22.108C25 C23.144C24 C42⋊14D6 C4⋊C4⋊26D6 C4⋊C4⋊28D6 ...
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 C42⋊5Q8 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 C24⋊13D4 C24⋊8Q8 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: C42⋊42D4 C4×C4⋊D4 C42⋊15D4 C42⋊16D4 C42⋊17D4 C42⋊19D4 C42⋊22D4 C42⋊23D4 ...
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
| 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 |
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
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