p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.407D4, C42.145C23, C22.25C4≀C2, C42.86(C2×C4), C4.4D4.8C4, (C22×D4).8C4, C8⋊C4⋊51C22, (C22×Q8).7C4, (C4×M4(2))⋊18C2, C4.5(C4.D4), (C22×C4).225D4, C42.6C4⋊33C2, C42.C22⋊7C2, (C2×C42).189C22, C23.177(C22⋊C4), C4.4D4.113C22, C2.27(C42⋊C22), C2.32(C2×C4≀C2), (C2×D4).20(C2×C4), (C2×Q8).20(C2×C4), (C2×C4).1173(C2×D4), (C2×C4.4D4).3C2, C2.11(C2×C4.D4), (C2×C4).139(C22×C4), (C22×C4).211(C2×C4), (C2×C4).177(C22⋊C4), C22.203(C2×C22⋊C4), SmallGroup(128,259)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.407D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 316 in 132 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C2×M4(2), C22×D4, C22×Q8, C42.C22, C4×M4(2), C42.6C4, C2×C4.4D4, C42.407D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C4≀C2, C2×C22⋊C4, C2×C4.D4, C2×C4≀C2, C42⋊C22, C42.407D4
►On 32 points
(1 26 18 14)(2 15 19 27)(3 28 20 16)(4 9 21 29)(5 30 22 10)(6 11 23 31)(7 32 24 12)(8 13 17 25)
(1 16 22 32)(2 13 23 29)(3 10 24 26)(4 15 17 31)(5 12 18 28)(6 9 19 25)(7 14 20 30)(8 11 21 27)
(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 11 16 21 22 27 32 8)(2 24 13 26 23 3 29 10)(4 5 15 12 17 18 31 28)(6 20 9 30 19 7 25 14)
G:=sub<Sym(32)| (1,26,18,14)(2,15,19,27)(3,28,20,16)(4,9,21,29)(5,30,22,10)(6,11,23,31)(7,32,24,12)(8,13,17,25), (1,16,22,32)(2,13,23,29)(3,10,24,26)(4,15,17,31)(5,12,18,28)(6,9,19,25)(7,14,20,30)(8,11,21,27), (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), (1,11,16,21,22,27,32,8)(2,24,13,26,23,3,29,10)(4,5,15,12,17,18,31,28)(6,20,9,30,19,7,25,14)>;
G:=Group( (1,26,18,14)(2,15,19,27)(3,28,20,16)(4,9,21,29)(5,30,22,10)(6,11,23,31)(7,32,24,12)(8,13,17,25), (1,16,22,32)(2,13,23,29)(3,10,24,26)(4,15,17,31)(5,12,18,28)(6,9,19,25)(7,14,20,30)(8,11,21,27), (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), (1,11,16,21,22,27,32,8)(2,24,13,26,23,3,29,10)(4,5,15,12,17,18,31,28)(6,20,9,30,19,7,25,14) );
G=PermutationGroup([[(1,26,18,14),(2,15,19,27),(3,28,20,16),(4,9,21,29),(5,30,22,10),(6,11,23,31),(7,32,24,12),(8,13,17,25)], [(1,16,22,32),(2,13,23,29),(3,10,24,26),(4,15,17,31),(5,12,18,28),(6,9,19,25),(7,14,20,30),(8,11,21,27)], [(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)], [(1,11,16,21,22,27,32,8),(2,24,13,26,23,3,29,10),(4,5,15,12,17,18,31,28),(6,20,9,30,19,7,25,14)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C4≀C2 | C4.D4 | C42⋊C22 |
| kernel | C42.407D4 | C42.C22 | C4×M4(2) | C42.6C4 | C2×C4.4D4 | C4.4D4 | C22×D4 | C22×Q8 | C42 | C22×C4 | C22 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.407D4 ►in GL6(𝔽17)
| 16 | 9 | 0 | 0 | 0 | 0 |
| 13 | 1 | 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 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 3 | 9 | 1 | 15 |
| 0 | 0 | 16 | 14 | 1 | 16 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 2 | 0 |
| 0 | 0 | 10 | 9 | 2 | 15 |
| 0 | 0 | 7 | 4 | 3 | 14 |
| 0 | 0 | 9 | 13 | 5 | 5 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 10 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 2 | 0 |
| 0 | 0 | 7 | 5 | 0 | 2 |
| 0 | 0 | 11 | 15 | 0 | 3 |
| 0 | 0 | 0 | 6 | 10 | 12 |
G:=sub<GL(6,GF(17))| [16,13,0,0,0,0,9,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,1,0,0,0,0,2,4,0,0,0,0,0,0,1,1,3,16,0,0,15,16,9,14,0,0,0,0,1,1,0,0,0,0,15,16],[0,7,0,0,0,0,3,0,0,0,0,0,0,0,0,10,7,9,0,0,14,9,4,13,0,0,2,2,3,5,0,0,0,15,14,5],[0,10,0,0,0,0,3,12,0,0,0,0,0,0,0,7,11,0,0,0,14,5,15,6,0,0,2,0,0,10,0,0,0,2,3,12] >;
C42.407D4 in GAP, Magma, Sage, TeX
C_4^2._{407}D_4 % in TeX
G:=Group("C4^2.407D4"); // GroupNames label
G:=SmallGroup(128,259);
// by ID
G=gap.SmallGroup(128,259);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,1123,1018,248,1971,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations