direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C22×C22⋊C4, C24⋊4C4, C25.2C2, C23.58D4, C22.5C24, C23.68C23, C24.35C22, (C23×C4)⋊2C2, C23⋊5(C2×C4), (C2×C4)⋊3C23, C2.1(C23×C4), C2.1(C22×D4), C22⋊2(C22×C4), C22.57(C2×D4), (C22×C4)⋊15C22, SmallGroup(64,193)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×C22⋊C4
G = < a,b,c,d,e | a2=b2=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 505 in 337 conjugacy classes, 169 normal (6 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2×C22⋊C4, C23×C4, C25, C22×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, C22×C22⋊C4
►On 32 points
(1 5)(2 6)(3 7)(4 8)(9 21)(10 22)(11 23)(12 24)(13 32)(14 29)(15 30)(16 31)(17 27)(18 28)(19 25)(20 26)
(1 9)(2 10)(3 11)(4 12)(5 21)(6 22)(7 23)(8 24)(13 20)(14 17)(15 18)(16 19)(25 31)(26 32)(27 29)(28 30)
(1 11)(2 28)(3 9)(4 26)(5 23)(6 18)(7 21)(8 20)(10 30)(12 32)(13 24)(14 19)(15 22)(16 17)(25 29)(27 31)
(1 31)(2 32)(3 29)(4 30)(5 16)(6 13)(7 14)(8 15)(9 25)(10 26)(11 27)(12 28)(17 23)(18 24)(19 21)(20 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)
G:=sub<Sym(32)| (1,5)(2,6)(3,7)(4,8)(9,21)(10,22)(11,23)(12,24)(13,32)(14,29)(15,30)(16,31)(17,27)(18,28)(19,25)(20,26), (1,9)(2,10)(3,11)(4,12)(5,21)(6,22)(7,23)(8,24)(13,20)(14,17)(15,18)(16,19)(25,31)(26,32)(27,29)(28,30), (1,11)(2,28)(3,9)(4,26)(5,23)(6,18)(7,21)(8,20)(10,30)(12,32)(13,24)(14,19)(15,22)(16,17)(25,29)(27,31), (1,31)(2,32)(3,29)(4,30)(5,16)(6,13)(7,14)(8,15)(9,25)(10,26)(11,27)(12,28)(17,23)(18,24)(19,21)(20,22), (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)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,21)(10,22)(11,23)(12,24)(13,32)(14,29)(15,30)(16,31)(17,27)(18,28)(19,25)(20,26), (1,9)(2,10)(3,11)(4,12)(5,21)(6,22)(7,23)(8,24)(13,20)(14,17)(15,18)(16,19)(25,31)(26,32)(27,29)(28,30), (1,11)(2,28)(3,9)(4,26)(5,23)(6,18)(7,21)(8,20)(10,30)(12,32)(13,24)(14,19)(15,22)(16,17)(25,29)(27,31), (1,31)(2,32)(3,29)(4,30)(5,16)(6,13)(7,14)(8,15)(9,25)(10,26)(11,27)(12,28)(17,23)(18,24)(19,21)(20,22), (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) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,21),(10,22),(11,23),(12,24),(13,32),(14,29),(15,30),(16,31),(17,27),(18,28),(19,25),(20,26)], [(1,9),(2,10),(3,11),(4,12),(5,21),(6,22),(7,23),(8,24),(13,20),(14,17),(15,18),(16,19),(25,31),(26,32),(27,29),(28,30)], [(1,11),(2,28),(3,9),(4,26),(5,23),(6,18),(7,21),(8,20),(10,30),(12,32),(13,24),(14,19),(15,22),(16,17),(25,29),(27,31)], [(1,31),(2,32),(3,29),(4,30),(5,16),(6,13),(7,14),(8,15),(9,25),(10,26),(11,27),(12,28),(17,23),(18,24),(19,21),(20,22)], [(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)]])
C22×C22⋊C4 is a maximal subgroup of
C24.17Q8 C23⋊2C42 C24.50D4 C24.5Q8 C24.52D4 C25.3C4 C24.68D4 C24.78D4 C23⋊C42 C24.90D4 C23.194C24 C24.91D4 C23.203C24 C23.224C24 C23.240C24 C23.304C24 C24.94D4 C24⋊8D4 C23.311C24 C24.95D4 C23.318C24 C23.324C24 C24⋊4Q8 C23.372C24 C23.380C24 C23.382C24 C24.96D4 C23.434C24 C23.439C24 C23.461C24 C24⋊10D4 C24.97D4 C24⋊5Q8 D4×C22×C4 C22.79C25
C22×C22⋊C4 is a maximal quotient of
C25.85C22 C23.179C24 C24.90D4 C23.191C24 C23.192C24 C24.542C23 C24.549C23 C23.223C24 C24.73(C2×C4) D4○(C22⋊C8) C23.C24 C23.4C24 M4(2).24C23 M4(2).25C23 C24.98D4 2+ 1+4⋊5C4 2- 1+4⋊4C4 2- 1+4⋊5C4
40 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 4A | ··· | 4P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4 | D4 |
| kernel | C22×C22⋊C4 | C2×C22⋊C4 | C23×C4 | C25 | C24 | C23 |
| # reps | 1 | 12 | 2 | 1 | 16 | 8 |
Matrix representation of C22×C22⋊C4 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 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 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4],[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],[3,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,1,0] >;
C22×C22⋊C4 in GAP, Magma, Sage, TeX
C_2^2\times C_2^2\rtimes C_4
% in TeX
G:=Group("C2^2xC2^2:C4"); // GroupNames label
G:=SmallGroup(64,193);
// by ID
G=gap.SmallGroup(64,193);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,192,217]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations