p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.266D4, C42.397C23, C4.1002+ (1+4), (C2×C4)⋊7SD16, C8⋊5D4⋊22C2, C4⋊C8⋊76C22, (C4×C8)⋊42C22, C4⋊Q8⋊70C22, C4⋊SD16⋊36C2, C4.4D8⋊27C2, C4.86(C2×SD16), (C4×Q8)⋊11C22, C22⋊SD16⋊30C2, C4⋊C4.147C23, C4.27(C8⋊C22), (C2×C4).406C24, (C2×C8).325C23, C23.690(C2×D4), (C22×C4).496D4, D4⋊C4⋊45C22, (C2×SD16)⋊42C22, (C2×D4).156C23, (C2×Q8).143C23, C22.25(C2×SD16), C2.23(C22×SD16), C42.12C4⋊43C2, C4⋊1D4.162C22, C22⋊C8.219C22, (C2×C42).873C22, C22.666(C22×D4), C22⋊Q8.191C22, (C22×C4).1077C23, (C22×D4).387C22, C23.37C23⋊16C2, C2.77(C22.29C24), (C2×C4).866(C2×D4), C2.55(C2×C8⋊C22), (C2×C4⋊1D4).26C2, SmallGroup(128,1940)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 604 in 244 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×22], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×8], D4 [×28], Q8 [×6], C23, C23 [×16], C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], SD16 [×8], C22×C4 [×3], C2×D4 [×4], C2×D4 [×22], C2×Q8 [×2], C2×Q8, C24 [×2], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊1D4 [×4], C4⋊1D4 [×2], C4⋊Q8 [×2], C2×SD16 [×8], C22×D4 [×2], C22×D4 [×2], C42.12C4, C22⋊SD16 [×4], C4⋊SD16 [×4], C4.4D8 [×2], C8⋊5D4 [×2], C2×C4⋊1D4, C23.37C23, C42.266D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C8⋊C22 [×2], C22×D4, 2+ (1+4) [×2], C22.29C24, C22×SD16, C2×C8⋊C22, C42.266D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c3 >
►On 32 points
(1 10 31 19)(2 11 32 20)(3 12 25 21)(4 13 26 22)(5 14 27 23)(6 15 28 24)(7 16 29 17)(8 9 30 18)
(1 3 5 7)(2 30 6 26)(4 32 8 28)(9 24 13 20)(10 12 14 16)(11 18 15 22)(17 19 21 23)(25 27 29 31)
(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 9 5 13)(2 12 6 16)(3 15 7 11)(4 10 8 14)(17 32 21 28)(18 27 22 31)(19 30 23 26)(20 25 24 29)
G:=sub<Sym(32)| (1,10,31,19)(2,11,32,20)(3,12,25,21)(4,13,26,22)(5,14,27,23)(6,15,28,24)(7,16,29,17)(8,9,30,18), (1,3,5,7)(2,30,6,26)(4,32,8,28)(9,24,13,20)(10,12,14,16)(11,18,15,22)(17,19,21,23)(25,27,29,31), (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), (1,9,5,13)(2,12,6,16)(3,15,7,11)(4,10,8,14)(17,32,21,28)(18,27,22,31)(19,30,23,26)(20,25,24,29)>;
G:=Group( (1,10,31,19)(2,11,32,20)(3,12,25,21)(4,13,26,22)(5,14,27,23)(6,15,28,24)(7,16,29,17)(8,9,30,18), (1,3,5,7)(2,30,6,26)(4,32,8,28)(9,24,13,20)(10,12,14,16)(11,18,15,22)(17,19,21,23)(25,27,29,31), (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), (1,9,5,13)(2,12,6,16)(3,15,7,11)(4,10,8,14)(17,32,21,28)(18,27,22,31)(19,30,23,26)(20,25,24,29) );
G=PermutationGroup([(1,10,31,19),(2,11,32,20),(3,12,25,21),(4,13,26,22),(5,14,27,23),(6,15,28,24),(7,16,29,17),(8,9,30,18)], [(1,3,5,7),(2,30,6,26),(4,32,8,28),(9,24,13,20),(10,12,14,16),(11,18,15,22),(17,19,21,23),(25,27,29,31)], [(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)], [(1,9,5,13),(2,12,6,16),(3,15,7,11),(4,10,8,14),(17,32,21,28),(18,27,22,31),(19,30,23,26),(20,25,24,29)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1,0,0,0],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | C8⋊C22 | 2+ (1+4) |
| kernel | C42.266D4 | C42.12C4 | C22⋊SD16 | C4⋊SD16 | C4.4D8 | C8⋊5D4 | C2×C4⋊1D4 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{266}D_4 % in TeX
G:=Group("C4^2.266D4"); // GroupNames label
G:=SmallGroup(128,1940);
// by ID
G=gap.SmallGroup(128,1940);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations