p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.691C23, C4.1662+ (1+4), (C2×D4)⋊9C8, D4⋊7(C2×C8), (C8×D4)⋊3C2, C23⋊3(C2×C8), (C4×C8)⋊4C22, C4⋊C8⋊95C22, D4○2(C22⋊C8), (C4×D4).32C4, C2.7(C23×C8), C4.19(C22×C8), C24.83(C2×C4), (C22×C8)⋊5C22, C22⋊C8⋊83C22, (C2×C4).669C24, (C2×C8).481C23, C42.220(C2×C4), (C22×D4).41C4, C22.2(C22×C8), (C4×D4).362C22, C2.3(Q8○M4(2)), C22.43(C23×C4), C42.12C4⋊23C2, (C23×C4).186C22, C23.149(C22×C4), (C2×C42).779C22, (C22×C4).1280C23, C2.4(C22.11C24), C4⋊C8○(C4⋊C8), (C2×C4)⋊4(C2×C8), C22⋊C8○(C4×D4), C4⋊C4○(C22⋊C8), (C2×C4×D4).75C2, (C2×C4⋊C4).75C4, (C2×D4)○(C22⋊C8), C4⋊C4.250(C2×C4), (C2×C22⋊C8)⋊17C2, C22⋊C8○(C22⋊C8), C22⋊C4○(C22⋊C8), (C2×D4).252(C2×C4), C22⋊C4.93(C2×C4), (C2×C22⋊C4).33C4, (C2×C4).499(C22×C4), (C22×C4).137(C2×C4), SmallGroup(128,1704)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 250 conjugacy classes, 174 normal (16 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C4 [×7], C22, C22 [×10], C22 [×18], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×17], D4 [×16], C23, C23 [×12], C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×8], C22×C4 [×3], C22×C4 [×10], C22×C4 [×4], C2×D4 [×12], C24 [×2], C4×C8 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×8], C23×C4 [×2], C22×D4, C2×C22⋊C8 [×4], C42.12C4 [×2], C8×D4 [×8], C2×C4×D4, C42.691C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, 2+ (1+4) [×2], C22.11C24, C23×C8, Q8○M4(2), C42.691C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=1, c2=b, e2=a2, ab=ba, ac=ca, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >
►On 32 points
(1 23 31 10)(2 24 32 11)(3 17 25 12)(4 18 26 13)(5 19 27 14)(6 20 28 15)(7 21 29 16)(8 22 30 9)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(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 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)
(1 14 31 19)(2 20 32 15)(3 16 25 21)(4 22 26 9)(5 10 27 23)(6 24 28 11)(7 12 29 17)(8 18 30 13)
G:=sub<Sym(32)| (1,23,31,10)(2,24,32,11)(3,17,25,12)(4,18,26,13)(5,19,27,14)(6,20,28,15)(7,21,29,16)(8,22,30,9), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (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), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30), (1,14,31,19)(2,20,32,15)(3,16,25,21)(4,22,26,9)(5,10,27,23)(6,24,28,11)(7,12,29,17)(8,18,30,13)>;
G:=Group( (1,23,31,10)(2,24,32,11)(3,17,25,12)(4,18,26,13)(5,19,27,14)(6,20,28,15)(7,21,29,16)(8,22,30,9), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (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), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30), (1,14,31,19)(2,20,32,15)(3,16,25,21)(4,22,26,9)(5,10,27,23)(6,24,28,11)(7,12,29,17)(8,18,30,13) );
G=PermutationGroup([(1,23,31,10),(2,24,32,11),(3,17,25,12),(4,18,26,13),(5,19,27,14),(6,20,28,15),(7,21,29,16),(8,22,30,9)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(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)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30)], [(1,14,31,19),(2,20,32,15),(3,16,25,21),(4,22,26,9),(5,10,27,23),(6,24,28,11),(7,12,29,17),(8,18,30,13)])
Matrix representation ►G ⊆ GL5(𝔽17)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,1,0] >;
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4V | 8A | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | 2+ (1+4) | Q8○M4(2) |
| kernel | C42.691C23 | C2×C22⋊C8 | C42.12C4 | C8×D4 | C2×C4×D4 | C2×C22⋊C4 | C2×C4⋊C4 | C4×D4 | C22×D4 | C2×D4 | C4 | C2 |
| # reps | 1 | 4 | 2 | 8 | 1 | 4 | 2 | 8 | 2 | 32 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{691}C_2^3 % in TeX
G:=Group("C4^2.691C2^3"); // GroupNames label
G:=SmallGroup(128,1704);
// by ID
G=gap.SmallGroup(128,1704);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=b,e^2=a^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=a^2*d>;
// generators/relations