p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.426D4, C8⋊C22⋊3C4, C4○D4.25D4, C4.118(C4×D4), C8.C22⋊3C4, C22.8(C4×D4), C42⋊6C4⋊4C2, (C2×D4).277D4, M4(2)⋊6(C2×C4), (C2×Q8).218D4, C22⋊C4.122D4, D4.8(C22⋊C4), C23.131(C2×D4), Q8.8(C22⋊C4), C4.190(C4⋊D4), M4(2)⋊4C4⋊7C2, C22.33C22≀C2, D8⋊C22.2C2, C22.3(C4⋊D4), C42⋊C22⋊15C2, (C22×C4).690C23, C23.C23⋊6C2, (C2×C42).289C22, C4.12(C22.D4), C42⋊C2.270C22, C2.45(C23.23D4), (C2×M4(2)).191C22, (C4×C4○D4)⋊1C2, (C2×C4≀C2)⋊17C2, (C2×D4)⋊11(C2×C4), (C2×Q8)⋊11(C2×C4), C4○D4.15(C2×C4), C4.22(C2×C22⋊C4), (C2×C4).1008(C2×D4), (C2×C4).12(C22×C4), (C2×C4).759(C4○D4), (C2×C4○D4).264C22, SmallGroup(128,638)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.426D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=a-1b-1, bc=cb, bd=db, dcd-1=a2b2c-1 >
Subgroups: 340 in 166 conjugacy classes, 54 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C23⋊C4, C4≀C2, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×M4(2), C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, C2×C4○D4, C42⋊6C4, M4(2)⋊4C4, C23.C23, C2×C4≀C2, C42⋊C22, C4×C4○D4, D8⋊C22, C42.426D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C42.426D4
►On 16 points - transitive group 16T225
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 3 2 4)(5 7 6 8)(9 10 11 12)(13 14 15 16)
(1 9 7 14)(2 11 8 16)(3 10 6 15)(4 12 5 13)
(1 12 2 10)(3 9 4 11)(5 14 6 16)(7 15 8 13)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,2,4)(5,7,6,8)(9,10,11,12)(13,14,15,16), (1,9,7,14)(2,11,8,16)(3,10,6,15)(4,12,5,13), (1,12,2,10)(3,9,4,11)(5,14,6,16)(7,15,8,13)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,2,4)(5,7,6,8)(9,10,11,12)(13,14,15,16), (1,9,7,14)(2,11,8,16)(3,10,6,15)(4,12,5,13), (1,12,2,10)(3,9,4,11)(5,14,6,16)(7,15,8,13) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3,2,4),(5,7,6,8),(9,10,11,12),(13,14,15,16)], [(1,9,7,14),(2,11,8,16),(3,10,6,15),(4,12,5,13)], [(1,12,2,10),(3,9,4,11),(5,14,6,16),(7,15,8,13)]])
G:=TransitiveGroup(16,225);
►On 16 points - transitive group 16T311
(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 16 15 14)
(1 9)(2 11)(3 10)(4 12)(5 15 6 13)(7 16 8 14)
(1 15 2 13)(3 16 4 14)(5 11 6 9)(7 12 8 10)
G:=sub<Sym(16)| (9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,16,15,14), (1,9)(2,11)(3,10)(4,12)(5,15,6,13)(7,16,8,14), (1,15,2,13)(3,16,4,14)(5,11,6,9)(7,12,8,10)>;
G:=Group( (9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,16,15,14), (1,9)(2,11)(3,10)(4,12)(5,15,6,13)(7,16,8,14), (1,15,2,13)(3,16,4,14)(5,11,6,9)(7,12,8,10) );
G=PermutationGroup([[(9,10,11,12),(13,14,15,16)], [(1,4,2,3),(5,8,6,7),(9,12,11,10),(13,16,15,14)], [(1,9),(2,11),(3,10),(4,12),(5,15,6,13),(7,16,8,14)], [(1,15,2,13),(3,16,4,14),(5,11,6,9),(7,12,8,10)]])
G:=TransitiveGroup(16,311);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D4 | D4 | D4 | C4○D4 | C42.426D4 |
| kernel | C42.426D4 | C42⋊6C4 | M4(2)⋊4C4 | C23.C23 | C2×C4≀C2 | C42⋊C22 | C4×C4○D4 | D8⋊C22 | C8⋊C22 | C8.C22 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 4 | 4 |
Matrix representation of C42.426D4 ►in GL4(𝔽5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [3,0,0,0,0,4,0,0,0,0,3,0,0,0,0,4],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,1,0,0,0,0,2,0,0,0,0,1,3,0,0,0],[0,4,0,0,1,0,0,0,0,0,0,1,0,0,4,0] >;
C42.426D4 in GAP, Magma, Sage, TeX
C_4^2._{426}D_4 % in TeX
G:=Group("C4^2.426D4"); // GroupNames label
G:=SmallGroup(128,638);
// by ID
G=gap.SmallGroup(128,638);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,521,1411,718,172,2028,1027]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^-1>;
// generators/relations