p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.404D4, C4⋊Q8⋊7C4, (C4×Q8)⋊2C4, C4.25C4≀C2, (C2×C4).55Q16, C42.71(C2×C4), (C2×C4).99SD16, C22.8(C2×Q16), (C22×C4).733D4, C23.493(C2×D4), C22.9(C2×SD16), C4.33(Q8⋊C4), C4○3(C23.31D4), C22⋊C8.164C22, (C22×C4).625C23, (C2×C42).174C22, C23.31D4.7C2, C22⋊Q8.136C22, C42.12C4.20C2, C23.37C23.5C2, C2.C42.502C22, C2.14(C23.C23), (C4×C4⋊C4).4C2, C4⋊C4.4(C2×C4), C2.20(C2×C4≀C2), (C2×Q8).5(C2×C4), C2.8(C2×Q8⋊C4), (C2×C4).1149(C2×D4), (C2×C4).88(C22⋊C4), (C2×C4).115(C22×C4), C22.179(C2×C22⋊C4), SmallGroup(128,235)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.404D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b, ab=ba, ac=ca, ad=da, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 228 in 120 conjugacy classes, 54 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C23.31D4, C4×C4⋊C4, C42.12C4, C23.37C23, C42.404D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, SD16, Q16, C22×C4, C2×D4, Q8⋊C4, C4≀C2, C2×C22⋊C4, C2×SD16, C2×Q16, C23.C23, C2×Q8⋊C4, C2×C4≀C2, C42.404D4
►On 32 points
(1 12 25 21)(2 13 26 22)(3 14 27 23)(4 15 28 24)(5 16 29 17)(6 9 30 18)(7 10 31 19)(8 11 32 20)
(1 27 5 31)(2 28 6 32)(3 29 7 25)(4 30 8 26)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(1 17 25 16)(2 11)(3 14 27 23)(4 18)(5 21 29 12)(6 15)(7 10 31 19)(8 22)(9 28)(13 32)(20 26)(24 30)
(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)| (1,12,25,21)(2,13,26,22)(3,14,27,23)(4,15,28,24)(5,16,29,17)(6,9,30,18)(7,10,31,19)(8,11,32,20), (1,27,5,31)(2,28,6,32)(3,29,7,25)(4,30,8,26)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (1,17,25,16)(2,11)(3,14,27,23)(4,18)(5,21,29,12)(6,15)(7,10,31,19)(8,22)(9,28)(13,32)(20,26)(24,30), (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( (1,12,25,21)(2,13,26,22)(3,14,27,23)(4,15,28,24)(5,16,29,17)(6,9,30,18)(7,10,31,19)(8,11,32,20), (1,27,5,31)(2,28,6,32)(3,29,7,25)(4,30,8,26)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (1,17,25,16)(2,11)(3,14,27,23)(4,18)(5,21,29,12)(6,15)(7,10,31,19)(8,22)(9,28)(13,32)(20,26)(24,30), (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([[(1,12,25,21),(2,13,26,22),(3,14,27,23),(4,15,28,24),(5,16,29,17),(6,9,30,18),(7,10,31,19),(8,11,32,20)], [(1,27,5,31),(2,28,6,32),(3,29,7,25),(4,30,8,26),(9,20,13,24),(10,21,14,17),(11,22,15,18),(12,23,16,19)], [(1,17,25,16),(2,11),(3,14,27,23),(4,18),(5,21,29,12),(6,15),(7,10,31,19),(8,22),(9,28),(13,32),(20,26),(24,30)], [(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)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4T | 4U | 4V | 4W | 4X | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | Q16 | C4≀C2 | C23.C23 |
| kernel | C42.404D4 | C23.31D4 | C4×C4⋊C4 | C42.12C4 | C23.37C23 | C4×Q8 | C4⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C42.404D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 15 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 10 | 0 | 0 |
| 12 | 10 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,1,0,0,15,16,0,0,0,0,4,0,0,0,0,4],[16,16,0,0,0,1,0,0,0,0,4,0,0,0,0,16],[0,12,0,0,10,10,0,0,0,0,0,4,0,0,16,0] >;
C42.404D4 in GAP, Magma, Sage, TeX
C_4^2._{404}D_4 % in TeX
G:=Group("C4^2.404D4"); // GroupNames label
G:=SmallGroup(128,235);
// by ID
G=gap.SmallGroup(128,235);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,520,1123,1018,248,1971]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations