p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.446D4, C42.325C23, C4⋊C8⋊4C22, D4.6(C2×D4), Q8.6(C2×D4), C4○D4.30D4, (C4×D4)⋊84C22, (C2×C8).15C23, (C4×Q8)⋊80C22, C4.71(C22×D4), D4.2D4⋊13C2, C4⋊C4.381C23, C4⋊M4(2)⋊6C2, (C2×C4).244C24, Q8.D4⋊13C2, (C2×Q16)⋊16C22, (C2×SD16)⋊8C22, (C2×D8).54C22, C23.656(C2×D4), (C22×C4).424D4, (C2×Q8).38C23, C4.106(C4⋊D4), Q8⋊C4⋊18C22, (C2×D4).387C23, C23.38D4⋊7C2, C23.37D4⋊7C2, D4⋊C4.21C22, C22.79(C4⋊D4), (C2×C42).813C22, (C22×C4).974C23, C22.504(C22×D4), C2.13(D8⋊C22), C4.4D4.127C22, (C22×D4).339C22, (C2×M4(2)).51C22, (C22×Q8).272C22, C42⋊C2.313C22, (C4×C4○D4)⋊8C2, C4.154(C2×C4○D4), C2.62(C2×C4⋊D4), (C2×C4).1214(C2×D4), (C2×C4.4D4)⋊39C2, (C2×C8⋊C22).10C2, (C2×C8.C22)⋊15C2, (C2×C4).275(C4○D4), (C2×C4○D4).300C22, SmallGroup(128,1772)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 500 in 250 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×16], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], D4 [×2], D4 [×11], Q8 [×2], Q8 [×7], C23, C23 [×9], C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×2], C24, D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C2×C42, C2×C22⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4.4D4 [×4], C4.4D4 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C8⋊C22 [×4], C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4, C23.37D4, C23.38D4, C4⋊M4(2), D4.2D4 [×4], Q8.D4 [×4], C4×C4○D4, C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C42.446D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D8⋊C22 [×2], C42.446D4
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=dad=a-1b2, cbc-1=dbd=b-1, dcd=b2c3 >
►On 32 points
(1 30 19 14)(2 11 20 27)(3 32 21 16)(4 13 22 29)(5 26 23 10)(6 15 24 31)(7 28 17 12)(8 9 18 25)
(1 17 5 21)(2 22 6 18)(3 19 7 23)(4 24 8 20)(9 27 13 31)(10 32 14 28)(11 29 15 25)(12 26 16 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)
(2 8)(3 7)(4 6)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 32)(17 21)(18 20)(22 24)
G:=sub<Sym(32)| (1,30,19,14)(2,11,20,27)(3,32,21,16)(4,13,22,29)(5,26,23,10)(6,15,24,31)(7,28,17,12)(8,9,18,25), (1,17,5,21)(2,22,6,18)(3,19,7,23)(4,24,8,20)(9,27,13,31)(10,32,14,28)(11,29,15,25)(12,26,16,30), (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), (2,8)(3,7)(4,6)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,32)(17,21)(18,20)(22,24)>;
G:=Group( (1,30,19,14)(2,11,20,27)(3,32,21,16)(4,13,22,29)(5,26,23,10)(6,15,24,31)(7,28,17,12)(8,9,18,25), (1,17,5,21)(2,22,6,18)(3,19,7,23)(4,24,8,20)(9,27,13,31)(10,32,14,28)(11,29,15,25)(12,26,16,30), (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), (2,8)(3,7)(4,6)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,32)(17,21)(18,20)(22,24) );
G=PermutationGroup([(1,30,19,14),(2,11,20,27),(3,32,21,16),(4,13,22,29),(5,26,23,10),(6,15,24,31),(7,28,17,12),(8,9,18,25)], [(1,17,5,21),(2,22,6,18),(3,19,7,23),(4,24,8,20),(9,27,13,31),(10,32,14,28),(11,29,15,25),(12,26,16,30)], [(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)], [(2,8),(3,7),(4,6),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,32),(17,21),(18,20),(22,24)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 4 |
| 0 | 0 | 4 | 13 | 13 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 16 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,4,0,4,0,0,9,13,13,13,0,0,0,0,0,13,0,0,0,0,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,16,0,0,0,1,0,0,0,0,15,16,16,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 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 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D8⋊C22 |
| kernel | C42.446D4 | C23.37D4 | C23.38D4 | C4⋊M4(2) | D4.2D4 | Q8.D4 | C4×C4○D4 | C2×C4.4D4 | C2×C8⋊C22 | C2×C8.C22 | C42 | C22×C4 | C4○D4 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{446}D_4 % in TeX
G:=Group("C4^2.446D4"); // GroupNames label
G:=SmallGroup(128,1772);
// by ID
G=gap.SmallGroup(128,1772);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2019,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=b^2*c^3>;
// generators/relations