p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.691C23, C4.1662+ 1+4, D4:7(C2xC8), (C2xD4):9C8, (C8xD4):3C2, C23:3(C2xC8), (C4xC8):4C22, C4:C8:95C22, D4o2(C22:C8), (C4xD4).32C4, C2.7(C23xC8), C4.19(C22xC8), C24.83(C2xC4), (C22xC8):5C22, C22:C8:83C22, (C2xC4).669C24, (C2xC8).481C23, C42.220(C2xC4), (C22xD4).41C4, C22.2(C22xC8), (C4xD4).362C22, C2.3(Q8oM4(2)), C22.43(C23xC4), C42.12C4:23C2, (C23xC4).186C22, C23.149(C22xC4), (C2xC42).779C22, (C22xC4).1280C23, C2.4(C22.11C24), C4:C8o(C4:C8), (C2xC4):4(C2xC8), C22:C8o(C4xD4), C4:C4o(C22:C8), (C2xC4xD4).75C2, (C2xC4:C4).75C4, (C2xD4)o(C22:C8), C4:C4.250(C2xC4), (C2xC22:C8):17C2, C22:C8o(C22:C8), C22:C4o(C22:C8), (C2xD4).252(C2xC4), C22:C4.93(C2xC4), (C2xC22:C4).33C4, (C2xC4).499(C22xC4), (C22xC4).137(C2xC4), SmallGroup(128,1704)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.691C23
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 >
Subgroups: 388 in 250 conjugacy classes, 174 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, C23, C23, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, C22xC4, C22xC4, C22xC4, C2xD4, C24, C4xC8, C22:C8, C4:C8, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C22xC8, C23xC4, C22xD4, C2xC22:C8, C42.12C4, C8xD4, C2xC4xD4, C42.691C23
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C2xC8, C22xC4, C24, C22xC8, C23xC4, 2+ 1+4, C22.11C24, C23xC8, Q8oM4(2), C42.691C23
►On 32 points
(1 23 31 12)(2 24 32 13)(3 17 25 14)(4 18 26 15)(5 19 27 16)(6 20 28 9)(7 21 29 10)(8 22 30 11)
(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 16 31 19)(2 20 32 9)(3 10 25 21)(4 22 26 11)(5 12 27 23)(6 24 28 13)(7 14 29 17)(8 18 30 15)
G:=sub<Sym(32)| (1,23,31,12)(2,24,32,13)(3,17,25,14)(4,18,26,15)(5,19,27,16)(6,20,28,9)(7,21,29,10)(8,22,30,11), (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), (1,16,31,19)(2,20,32,9)(3,10,25,21)(4,22,26,11)(5,12,27,23)(6,24,28,13)(7,14,29,17)(8,18,30,15)>;
G:=Group( (1,23,31,12)(2,24,32,13)(3,17,25,14)(4,18,26,15)(5,19,27,16)(6,20,28,9)(7,21,29,10)(8,22,30,11), (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), (1,16,31,19)(2,20,32,9)(3,10,25,21)(4,22,26,11)(5,12,27,23)(6,24,28,13)(7,14,29,17)(8,18,30,15) );
G=PermutationGroup([[(1,23,31,12),(2,24,32,13),(3,17,25,14),(4,18,26,15),(5,19,27,16),(6,20,28,9),(7,21,29,10),(8,22,30,11)], [(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)], [(1,16,31,19),(2,20,32,9),(3,10,25,21),(4,22,26,11),(5,12,27,23),(6,24,28,13),(7,14,29,17),(8,18,30,15)]])
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 | Q8oM4(2) |
| kernel | C42.691C23 | C2xC22:C8 | C42.12C4 | C8xD4 | C2xC4xD4 | C2xC22:C4 | C2xC4:C4 | C4xD4 | C22xD4 | C2xD4 | C4 | C2 |
| # reps | 1 | 4 | 2 | 8 | 1 | 4 | 2 | 8 | 2 | 32 | 2 | 2 |
Matrix representation of C42.691C23 ►in 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] >;
C42.691C23 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