p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.107D4, (C2×C8).201D4, C4.127(C4×D4), D4⋊C4⋊12C4, Q8⋊C4⋊12C4, C42⋊6C4⋊29C2, C22.172(C4×D4), C2.17(C8.26D4), C4.196(C4⋊D4), C4.C42⋊21C2, C4.87(C4.4D4), C4.12(C42⋊C2), C23.208(C4○D4), (C2×C42).304C22, (C22×C8).399C22, C42.6C22⋊23C2, (C22×C4).1389C23, C42⋊C2.39C22, C22.2(C42⋊2C2), C23.24D4.10C2, (C2×M4(2)).201C22, C2.23(C24.C22), C22.31(C22.D4), C4⋊C4.86(C2×C4), (C2×C8⋊C4)⋊25C2, (C2×C4≀C2).13C2, (C2×C8).148(C2×C4), (C2×Q8).87(C2×C4), (C2×D4).102(C2×C4), (C2×C4).1346(C2×D4), (C2×C4).584(C4○D4), (C2×C4).407(C22×C4), (C2×C4○D4).36C22, (C22×C8)⋊C2.17C2, SmallGroup(128,670)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.107D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=a-1b, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=a2b2c3 >
Subgroups: 212 in 111 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, C8⋊C4, 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×C8⋊C4, (C22×C8)⋊C2, C23.24D4, C2×C4≀C2, C42.6C22, C42.107D4
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, C8.26D4, C42.107D4
►On 32 points
(2 26 6 30)(4 28 8 32)(9 13)(10 22 14 18)(11 15)(12 24 16 20)(17 21)(19 23)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(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 18 25 14 5 22 29 10)(2 21 26 9 6 17 30 13)(3 20 27 16 7 24 31 12)(4 23 28 11 8 19 32 15)
G:=sub<Sym(32)| (2,26,6,30)(4,28,8,32)(9,13)(10,22,14,18)(11,15)(12,24,16,20)(17,21)(19,23), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (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), (1,18,25,14,5,22,29,10)(2,21,26,9,6,17,30,13)(3,20,27,16,7,24,31,12)(4,23,28,11,8,19,32,15)>;
G:=Group( (2,26,6,30)(4,28,8,32)(9,13)(10,22,14,18)(11,15)(12,24,16,20)(17,21)(19,23), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (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), (1,18,25,14,5,22,29,10)(2,21,26,9,6,17,30,13)(3,20,27,16,7,24,31,12)(4,23,28,11,8,19,32,15) );
G=PermutationGroup([[(2,26,6,30),(4,28,8,32),(9,13),(10,22,14,18),(11,15),(12,24,16,20),(17,21),(19,23)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24)], [(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)], [(1,18,25,14,5,22,29,10),(2,21,26,9,6,17,30,13),(3,20,27,16,7,24,31,12),(4,23,28,11,8,19,32,15)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D4 | C4○D4 | C8.26D4 |
| kernel | C42.107D4 | C42⋊6C4 | C4.C42 | C2×C8⋊C4 | (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 | 4 |
Matrix representation of C42.107D4 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 10 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 14 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 8 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 8 | 0 | 13 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 8 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,10,0,4,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,14,16,0,0,0,0,0,0,0,0,0,8,0,0,8,0,16,0,0,0,0,4,0,13,0,0,8,0,0,0],[8,11,0,0,0,0,8,9,0,0,0,0,0,0,4,0,8,0,0,0,0,0,0,13,0,0,6,0,13,0,0,0,0,1,0,0] >;
C42.107D4 in GAP, Magma, Sage, TeX
C_4^2._{107}D_4 % in TeX
G:=Group("C4^2.107D4"); // GroupNames label
G:=SmallGroup(128,670);
// by ID
G=gap.SmallGroup(128,670);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,58,2019,248,2804,172,124]);
// 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=a^-1*b,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