p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4○C2≀C4, C2≀C4⋊7C2, (C23×C4)⋊9C4, C22≀C2⋊6C4, C23.3(C2×D4), C42⋊C2⋊6C4, (C2×D4).125D4, C24.32(C2×C4), (C2×Q8).114D4, (C22×C4).90D4, C4.29(C23⋊C4), (C2×D4).14C23, C4○(C23.D4), C23.D4⋊7C2, C22.D4⋊6C4, C23⋊C4.7C22, C23.53(C22×C4), C4.D4.8C22, C22≀C2.20C22, C23.19(C22⋊C4), C22.19C24.9C2, C23.C23⋊12C2, C22.D4.24C22, M4(2).8C22⋊14C2, (C2×C4).3(C2×D4), C22⋊C4.1(C2×C4), C2.32(C2×C23⋊C4), (C2×D4).123(C2×C4), (C22×C4).78(C2×C4), (C2×C4○D4).71C22, C22.56(C2×C22⋊C4), (C2×C4).364(C22⋊C4), SmallGroup(128,852)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4○C2≀C4
G = < a,b,c,d,e,f | a4=b2=c2=d2=f4=1, e1=a2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=a2bcd, cd=dc, ce=ec, fcf-1=a2cd, de=ed, fdf-1=a2d, ef=fe >
Subgroups: 324 in 132 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C23⋊C4, C23⋊C4, C4.D4, C4.10D4, C42⋊C2, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C2×M4(2), C23×C4, C2×C4○D4, C2≀C4, C23.D4, C23.C23, M4(2).8C22, C22.19C24, C4○C2≀C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C4○C2≀C4
Character table of C4○C2≀C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -i | -i | i | i | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | i | i | -i | -i | 1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -i | -i | i | i | 1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | i | i | -i | -i | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | -2 | -2 | -2 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 0 | -2 | -2 | -2 | 0 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ22 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ23 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 2 | 4i | -4i | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 2 | 0 | 0 | -2 | 4i | -4i | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 2 | -4i | 4i | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 2 | 0 | 0 | -2 | -4i | 4i | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T243
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 14 12 8)(2 15 9 5)(3 16 10 6)(4 13 11 7)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (5,7)(6,8)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,14,12,8)(2,15,9,5)(3,16,10,6)(4,13,11,7)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (5,7)(6,8)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,14,12,8)(2,15,9,5)(3,16,10,6)(4,13,11,7) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,11),(10,12)], [(1,11),(2,12),(3,9),(4,10),(5,14),(6,15),(7,16),(8,13)], [(5,7),(6,8),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,14,12,8),(2,15,9,5),(3,16,10,6),(4,13,11,7)]])
G:=TransitiveGroup(16,243);
►On 16 points - transitive group 16T280
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(5 16)(6 13)(7 14)(8 15)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 16)(2 13)(3 14)(4 15)(5 11 7 9)(6 12 8 10)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,16)(6,13)(7,14)(8,15), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15), (5,7)(6,8)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,16)(2,13)(3,14)(4,15)(5,11,7,9)(6,12,8,10)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,16)(6,13)(7,14)(8,15), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15), (5,7)(6,8)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,16)(2,13)(3,14)(4,15)(5,11,7,9)(6,12,8,10) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(5,16),(6,13),(7,14),(8,15)], [(1,11),(2,12),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15)], [(5,7),(6,8),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,16),(2,13),(3,14),(4,15),(5,11,7,9),(6,12,8,10)]])
G:=TransitiveGroup(16,280);
►On 16 points - transitive group 16T292
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(5 7)(6 8)
(5 7)(6 8)(9 11)(10 12)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 5 11 14)(2 6 12 15)(3 7 9 16)(4 8 10 13)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,7)(6,8), (5,7)(6,8)(9,11)(10,12), (5,7)(6,8)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,5,11,14)(2,6,12,15)(3,7,9,16)(4,8,10,13)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (5,7)(6,8), (5,7)(6,8)(9,11)(10,12), (5,7)(6,8)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,5,11,14)(2,6,12,15)(3,7,9,16)(4,8,10,13) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(5,7),(6,8)], [(5,7),(6,8),(9,11),(10,12)], [(5,7),(6,8),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,5,11,14),(2,6,12,15),(3,7,9,16),(4,8,10,13)]])
G:=TransitiveGroup(16,292);
Matrix representation of C4○C2≀C4 ►in GL4(𝔽5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,0,1,0,3,0,0,0,0,0,0,1,0,2,0,0] >;
C4○C2≀C4 in GAP, Magma, Sage, TeX
C_4\circ C_2\wr C_4
% in TeX
G:=Group("C4oC2wrC4"); // GroupNames label
G:=SmallGroup(128,852);
// by ID
G=gap.SmallGroup(128,852);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,851,375,4037]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=c^2=d^2=f^4=1,e^1=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^2*b*c*d,c*d=d*c,c*e=e*c,f*c*f^-1=a^2*c*d,d*e=e*d,f*d*f^-1=a^2*d,e*f=f*e>;
// generators/relations
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