direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C22.D4, C23.49D4, C22.18C24, C23.35C23, C24.31C22, (C23×C4)⋊4C2, C4⋊C4⋊11C22, C2.7(C22×D4), (C2×C4).12C23, C22.18(C2×D4), C22⋊C4⋊14C22, (C22×C4)⋊17C22, (C22×D4).10C2, (C2×D4).60C22, C22.31(C4○D4), (C2×C4⋊C4)⋊16C2, C2.7(C2×C4○D4), (C2×C22⋊C4)⋊10C2, SmallGroup(64,205)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22.D4
G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 265 in 171 conjugacy classes, 89 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C2×C22.D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, C2×C22.D4
Character table of C2×C22.D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 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 | -2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 7)(2 8)(3 5)(4 6)(9 32)(10 29)(11 30)(12 31)(13 20)(14 17)(15 18)(16 19)(21 27)(22 28)(23 25)(24 26)
(1 28)(2 11)(3 26)(4 9)(5 24)(6 32)(7 22)(8 30)(10 13)(12 15)(14 25)(16 27)(17 23)(18 31)(19 21)(20 29)
(1 13)(2 14)(3 15)(4 16)(5 18)(6 19)(7 20)(8 17)(9 27)(10 28)(11 25)(12 26)(21 32)(22 29)(23 30)(24 31)
(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 5)(2 17)(3 7)(4 19)(6 16)(8 14)(9 32)(10 24)(11 30)(12 22)(13 18)(15 20)(21 27)(23 25)(26 29)(28 31)
G:=sub<Sym(32)| (1,7)(2,8)(3,5)(4,6)(9,32)(10,29)(11,30)(12,31)(13,20)(14,17)(15,18)(16,19)(21,27)(22,28)(23,25)(24,26), (1,28)(2,11)(3,26)(4,9)(5,24)(6,32)(7,22)(8,30)(10,13)(12,15)(14,25)(16,27)(17,23)(18,31)(19,21)(20,29), (1,13)(2,14)(3,15)(4,16)(5,18)(6,19)(7,20)(8,17)(9,27)(10,28)(11,25)(12,26)(21,32)(22,29)(23,30)(24,31), (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), (1,5)(2,17)(3,7)(4,19)(6,16)(8,14)(9,32)(10,24)(11,30)(12,22)(13,18)(15,20)(21,27)(23,25)(26,29)(28,31)>;
G:=Group( (1,7)(2,8)(3,5)(4,6)(9,32)(10,29)(11,30)(12,31)(13,20)(14,17)(15,18)(16,19)(21,27)(22,28)(23,25)(24,26), (1,28)(2,11)(3,26)(4,9)(5,24)(6,32)(7,22)(8,30)(10,13)(12,15)(14,25)(16,27)(17,23)(18,31)(19,21)(20,29), (1,13)(2,14)(3,15)(4,16)(5,18)(6,19)(7,20)(8,17)(9,27)(10,28)(11,25)(12,26)(21,32)(22,29)(23,30)(24,31), (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), (1,5)(2,17)(3,7)(4,19)(6,16)(8,14)(9,32)(10,24)(11,30)(12,22)(13,18)(15,20)(21,27)(23,25)(26,29)(28,31) );
G=PermutationGroup([[(1,7),(2,8),(3,5),(4,6),(9,32),(10,29),(11,30),(12,31),(13,20),(14,17),(15,18),(16,19),(21,27),(22,28),(23,25),(24,26)], [(1,28),(2,11),(3,26),(4,9),(5,24),(6,32),(7,22),(8,30),(10,13),(12,15),(14,25),(16,27),(17,23),(18,31),(19,21),(20,29)], [(1,13),(2,14),(3,15),(4,16),(5,18),(6,19),(7,20),(8,17),(9,27),(10,28),(11,25),(12,26),(21,32),(22,29),(23,30),(24,31)], [(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)], [(1,5),(2,17),(3,7),(4,19),(6,16),(8,14),(9,32),(10,24),(11,30),(12,22),(13,18),(15,20),(21,27),(23,25),(26,29),(28,31)]])
C2×C22.D4 is a maximal subgroup of
C23⋊C8⋊C2 C24.26D4 C24.174C23 C24.31D4 C24.195C23 C24.204C23 C23.241C24 C24.223C23 C24.225C23 C24.94D4 C24.243C23 C23.311C24 C24.95D4 C23.313C24 C23.318C24 C24.563C23 C23.322C24 C24.258C23 C24.262C23 C24.269C23 C23.345C24 C24.276C23 C23.356C24 C24.278C23 C24.279C23 C24.282C23 C23.364C24 C24.289C23 C24.290C23 C24.299C23 C23.388C24 C23.398C24 C23.401C24 C23.434C24 C23.439C24 C24.326C23 C24.327C23 C23.457C24 C23.458C24 C23.502C24 C24⋊9D4 C23.514C24 C24.587C23 C24.589C23 C23.530C24 C24.377C23 C24.378C23 C23.571C24 C23.572C24 C23.573C24 C23.574C24 C23.578C24 C23.580C24 C23.581C24 C24.389C23 C24.394C23 C23.593C24 C24.401C23 C23.595C24 C24.403C23 C23.597C24 C24.407C23 C23.603C24 C23.605C24 C23.606C24 C23.607C24 C23.608C24 C24.411C23 C24.412C23 C23.617C24 C23.618C24 C23.624C24 C24.459C23 C23.714C24 C24.166D4 C23.753C24 C24.598C23 C22.74C25 C22.80C25 C22.102C25 C22.122C25 C22.123C25 C22.124C25
C2×C22.D4 is a maximal quotient of
C23.295C24 C24.94D4 C24.95D4 C23.318C24 C24.563C23 C24.254C23 C23.321C24 C23.322C24 C23.323C24 C24.269C23 C23.344C24 C23.345C24 C23.346C24 C24.271C23 C23.348C24 C23.382C24 C24.96D4 C24.576C23 C23.385C24 C24.299C23 C24.300C23 C23.398C24 C24.308C23 C23.400C24 C23.401C24 C23.402C24 C24.579C23 C23.404C24 C24.97D4 C24.589C23 C23.524C24 C23.525C24 C24.166D4 C23.753C24 C24.598C23 C24.599C23 C24.115D4 C24.183D4 C24.116D4 C24.117D4 C24.118D4 (C2×D4).301D4 (C2×D4).302D4 (C2×D4).303D4 (C2×D4).304D4
Matrix representation of C2×C22.D4 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 1 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,2,0,0,0,0,1,2,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;
C2×C22.D4 in GAP, Magma, Sage, TeX
C_2\times C_2^2.D_4
% in TeX
G:=Group("C2xC2^2.D4"); // GroupNames label
G:=SmallGroup(64,205);
// by ID
G=gap.SmallGroup(64,205);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,650,86]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations
Export