p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.428D4, D4⋊C4⋊2C4, Q8⋊C4⋊2C4, C4.126(C4×D4), (C2×C8).226D4, C2.17(C8○D8), C42⋊6C4⋊28C2, C22.171(C4×D4), C4.195(C4⋊D4), C4.C42⋊20C2, C4.86(C4.4D4), C4.11(C42⋊C2), C23.207(C4○D4), (C22×C8).485C22, C23.24D4.1C2, (C22×C4).1388C23, C42⋊C2.38C22, C42.6C22⋊22C2, (C2×C42).1066C22, C22.1(C42⋊2C2), (C2×M4(2)).200C22, C2.22(C24.C22), C22.30(C22.D4), (C2×C4×C8)⋊11C2, C4⋊C4.85(C2×C4), (C2×C4≀C2).12C2, (C2×C8).161(C2×C4), (C2×Q8).86(C2×C4), (C2×D4).101(C2×C4), (C2×C4).1345(C2×D4), (C2×C4).583(C4○D4), (C2×C4).406(C22×C4), (C2×C4○D4).35C22, (C22×C8)⋊C2.16C2, SmallGroup(128,669)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.428D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b-1, bc=cb, bd=db, dcd-1=a2b2c3 >
Subgroups: 212 in 115 conjugacy classes, 48 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4≀C2, C4⋊C8, C2×C42, C42⋊C2, C22×C8, C2×M4(2), C2×C4○D4, C42⋊6C4, C4.C42, C2×C4×C8, (C22×C8)⋊C2, C23.24D4, C2×C4≀C2, C42.6C22, C42.428D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C24.C22, C8○D8, C42.428D4
►On 32 points
(1 7 5 3)(2 27)(4 29)(6 31)(8 25)(9 18)(10 16 14 12)(11 20)(13 22)(15 24)(17 23 21 19)(26 32 30 28)
(1 28 5 32)(2 29 6 25)(3 30 7 26)(4 31 8 27)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(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 22 28 15 5 18 32 11)(2 21 29 14 6 17 25 10)(3 24 30 9 7 20 26 13)(4 23 31 16 8 19 27 12)
G:=sub<Sym(32)| (1,7,5,3)(2,27)(4,29)(6,31)(8,25)(9,18)(10,16,14,12)(11,20)(13,22)(15,24)(17,23,21,19)(26,32,30,28), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (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), (1,22,28,15,5,18,32,11)(2,21,29,14,6,17,25,10)(3,24,30,9,7,20,26,13)(4,23,31,16,8,19,27,12)>;
G:=Group( (1,7,5,3)(2,27)(4,29)(6,31)(8,25)(9,18)(10,16,14,12)(11,20)(13,22)(15,24)(17,23,21,19)(26,32,30,28), (1,28,5,32)(2,29,6,25)(3,30,7,26)(4,31,8,27)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (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), (1,22,28,15,5,18,32,11)(2,21,29,14,6,17,25,10)(3,24,30,9,7,20,26,13)(4,23,31,16,8,19,27,12) );
G=PermutationGroup([[(1,7,5,3),(2,27),(4,29),(6,31),(8,25),(9,18),(10,16,14,12),(11,20),(13,22),(15,24),(17,23,21,19),(26,32,30,28)], [(1,28,5,32),(2,29,6,25),(3,30,7,26),(4,31,8,27),(9,20,13,24),(10,21,14,17),(11,22,15,18),(12,23,16,19)], [(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)], [(1,22,28,15,5,18,32,11),(2,21,29,14,6,17,25,10),(3,24,30,9,7,20,26,13),(4,23,31,16,8,19,27,12)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 8A | ··· | 8P | 8Q | 8R | 8S | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | C4○D4 | C8○D8 |
| kernel | C42.428D4 | C42⋊6C4 | C4.C42 | C2×C4×C8 | (C22×C8)⋊C2 | C23.24D4 | C2×C4≀C2 | C42.6C22 | D4⋊C4 | Q8⋊C4 | C42 | C2×C8 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 6 | 2 | 16 |
Matrix representation of C42.428D4 ►in GL4(𝔽17) generated by
| 4 | 15 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 12 | 3 | 0 | 0 |
| 13 | 5 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 7 | 12 | 0 | 0 |
| 9 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,15,16,0,0,0,0,1,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[12,13,0,0,3,5,0,0,0,0,0,13,0,0,4,0],[7,9,0,0,12,10,0,0,0,0,0,16,0,0,1,0] >;
C42.428D4 in GAP, Magma, Sage, TeX
C_4^2._{428}D_4 % in TeX
G:=Group("C4^2.428D4"); // GroupNames label
G:=SmallGroup(128,669);
// by ID
G=gap.SmallGroup(128,669);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,58,2019,248,2804,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,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^3>;
// generators/relations