p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊4C22, C22.11C24, C23.31C23, C24.11C22, C2.12+ 1+4, D4⋊7(C2×C4), (C4×D4)⋊5C2, (C2×D4)⋊11C4, C23⋊3(C2×C4), D4○(C22⋊C4), C4⋊C4⋊20C22, C2.7(C23×C4), C42⋊C2⋊6C2, (C2×C4).50C23, C4.19(C22×C4), (C22×C4)⋊3C22, (C22×D4).9C2, C22⋊C4⋊18C22, (C2×D4).77C22, C22.2(C22×C4), C4⋊C4○(C4⋊C4), (C2×C4)⋊4(C2×C4), (C2×C22⋊C4)⋊5C2, C22⋊C4○(C22⋊C4), SmallGroup(64,199)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22.11C24
G = < a,b,c,d,e,f | a2=b2=d2=e2=f2=1, c2=b, ab=ba, dcd=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ce=ec, df=fd, ef=fe >
Subgroups: 257 in 169 conjugacy classes, 121 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C22.11C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, C22.11C24
►On 16 points - transitive group 16T68
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6)(2 10)(3 8)(4 12)(5 14)(7 16)(9 15)(11 13)
(1 3)(2 4)(5 10)(6 11)(7 12)(8 9)(13 15)(14 16)
(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)
G:=sub<Sym(16)| (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,10)(3,8)(4,12)(5,14)(7,16)(9,15)(11,13), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15)>;
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,10)(3,8)(4,12)(5,14)(7,16)(9,15)(11,13), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15) );
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6),(2,10),(3,8),(4,12),(5,14),(7,16),(9,15),(11,13)], [(1,3),(2,4),(5,10),(6,11),(7,12),(8,9),(13,15),(14,16)], [(1,3),(2,14),(4,16),(5,10),(6,8),(7,12),(9,11),(13,15)]])
G:=TransitiveGroup(16,68);
C22.11C24 is a maximal subgroup of
C24.C23 C24.6(C2×C4) C42.5D4 C23.C24 C42.275C23 C42.277C23 C42.278C23 C42.14C23 C42.15C23 C42.20C23 C42.352C23 C42.356C23 C42.357C23 C22.14C25 C4×2+ 1+4 C22.73C25 C22.74C25 C22.79C25 C22.80C25 C22.83C25 C22.99C25 C22.102C25 C22.110C25 D10.C24
C24.D2p: C24.21D4 C24.22D4 C24.23D4 C24.24D4 C25.C22 C24.26D4 C42⋊D4 C24.28D4 ...
C2p.2+ 1+4: C22.90C25 C22.94C25 C22.97C25 C22.103C25 C22.108C25 C42⋊9D6 C42⋊13D6 C42⋊7D10 ...
C22.11C24 is a maximal quotient of
C23⋊C42 C24.524C23 D4⋊4C42 C23.191C24 C23.194C24 C23.195C24 C24.545C23 C24.547C23 C23.201C24 C23.203C24 C24.198C23 C23.211C24 C42⋊4Q8 C23.214C24 C23.215C24 C24.204C23 C24.205C23 D4×C22⋊C4 C23.224C24 C23.227C24 C23.231C24 C23.235C24 C24.212C23 C23.240C24 C24.217C23 C24.218C23 C24.219C23 C23.250C24 C23.251C24 C24.221C23 C24.223C23 C23.257C24 C24.225C23 C23.259C24 C24.227C23 C23.261C24 C23.262C24 C23.263C24 C42.691C23 C23⋊3M4(2) D4⋊7M4(2) C42.693C23 C42.297C23 C42.298C23 C42.299C23 C42.694C23 C42.300C23 C42.301C23 D10.C24
C42⋊D2p: C42⋊13D4 C42⋊14D4 C42⋊9D6 C42⋊13D6 C42⋊7D10 C42⋊11D10 C42⋊7D14 C42⋊11D14 ...
C24.D2p: C24.90D4 C24.91D4 C24.35D6 C24.49D6 C24.24D10 C24.38D10 C24.24D14 C24.38D14 ...
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 4A | ··· | 4T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 |
| type | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C4 | 2+ 1+4 |
| kernel | C22.11C24 | C2×C22⋊C4 | C42⋊C2 | C4×D4 | C22×D4 | C2×D4 | C2 |
| # reps | 1 | 4 | 2 | 8 | 1 | 16 | 2 |
Matrix representation of C22.11C24 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,4,0,0,0,0,3,1,0,0,0,0,0,1,0,0,0,0,2,4],[4,0,0,0,0,0,1,4,0,0,0,0,4,0,0,0,0,0,1,4,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4] >;
C22.11C24 in GAP, Magma, Sage, TeX
C_2^2._{11}C_2^4 % in TeX
G:=Group("C2^2.11C2^4"); // GroupNames label
G:=SmallGroup(64,199);
// by ID
G=gap.SmallGroup(64,199);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,188,579]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=d^2=e^2=f^2=1,c^2=b,a*b=b*a,d*c*d=f*c*f=a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*e=e*c,d*f=f*d,e*f=f*e>;
// generators/relations