p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.269D4, C42.730C23, C4.1032+ 1+4, C4:C8:67C22, (C4xC8):16C22, Q8:D4:31C2, C22:D8.2C2, D4.2D4:2C2, C8.12D4:3C2, (C4xD4):12C22, C22:Q16:6C2, Q8.D4:2C2, (C2xQ16):5C22, (C4xQ8):12C22, C22:SD16:31C2, C4:C4.150C23, (C2xC8).327C23, (C2xC4).409C24, Q8:C4:5C22, (C2xD8).24C22, C23.692(C2xD4), (C22xC4).172D4, (C2xSD16):43C22, (C2xD4).158C23, C22.33(C4oD8), C4.4D4:60C22, (C2xQ8).146C23, C42.C2:37C22, C42.12C4:35C2, C4:D4.189C22, C22:C8.179C22, (C2xC42).876C22, C22.669(C22xD4), C22:Q8.194C22, D4:C4.107C22, C2.54(D8:C22), (C22xC4).1080C23, C42.78C22:7C2, C23.36C23:9C2, (C22xD4).388C22, (C22xQ8).321C22, C2.80(C22.29C24), C2.43(C2xC4oD8), (C2xC4).539(C2xD4), (C2xC4.4D4):43C2, SmallGroup(128,1943)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.269D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=b2c3 >
Subgroups: 452 in 205 conjugacy classes, 86 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C42, C22:C4, C4:C4, C4:C4, C2xC8, D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C24, C4xC8, C22:C8, D4:C4, Q8:C4, C4:C8, C2xC42, C2xC22:C4, C42:C2, C4xD4, C4xD4, C4xQ8, C4:D4, C22:Q8, C22.D4, C4.4D4, C4.4D4, C4.4D4, C42.C2, C42:2C2, C2xD8, C2xSD16, C2xQ16, C22xD4, C22xQ8, C42.12C4, C22:D8, Q8:D4, C22:SD16, C22:Q16, D4.2D4, Q8.D4, C42.78C22, C8.12D4, C2xC4.4D4, C23.36C23, C42.269D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4oD8, C22xD4, 2+ 1+4, C22.29C24, C2xC4oD8, D8:C22, C42.269D4
►On 32 points
(1 14 29 24)(2 15 30 17)(3 16 31 18)(4 9 32 19)(5 10 25 20)(6 11 26 21)(7 12 27 22)(8 13 28 23)
(1 7 5 3)(2 28 6 32)(4 30 8 26)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 24 22 20)(25 31 29 27)
(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 2)(3 8)(4 7)(5 6)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(25 26)(27 32)(28 31)(29 30)
G:=sub<Sym(32)| (1,14,29,24)(2,15,30,17)(3,16,31,18)(4,9,32,19)(5,10,25,20)(6,11,26,21)(7,12,27,22)(8,13,28,23), (1,7,5,3)(2,28,6,32)(4,30,8,26)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,24,22,20)(25,31,29,27), (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), (1,2)(3,8)(4,7)(5,6)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(25,26)(27,32)(28,31)(29,30)>;
G:=Group( (1,14,29,24)(2,15,30,17)(3,16,31,18)(4,9,32,19)(5,10,25,20)(6,11,26,21)(7,12,27,22)(8,13,28,23), (1,7,5,3)(2,28,6,32)(4,30,8,26)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,24,22,20)(25,31,29,27), (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), (1,2)(3,8)(4,7)(5,6)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(25,26)(27,32)(28,31)(29,30) );
G=PermutationGroup([[(1,14,29,24),(2,15,30,17),(3,16,31,18),(4,9,32,19),(5,10,25,20),(6,11,26,21),(7,12,27,22),(8,13,28,23)], [(1,7,5,3),(2,28,6,32),(4,30,8,26),(9,17,13,21),(10,16,14,12),(11,19,15,23),(18,24,22,20),(25,31,29,27)], [(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)], [(1,2),(3,8),(4,7),(5,6),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(25,26),(27,32),(28,31),(29,30)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4O | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4oD8 | 2+ 1+4 | D8:C22 |
| kernel | C42.269D4 | C42.12C4 | C22:D8 | Q8:D4 | C22:SD16 | C22:Q16 | D4.2D4 | Q8.D4 | C42.78C22 | C8.12D4 | C2xC4.4D4 | C23.36C23 | C42 | C22xC4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.269D4 ►in GL6(F17)
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 15 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 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 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 16 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 14 | 3 | 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))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,0,15,4,0,0,0,0,0,0,4,0,0,0,0,0,2,13],[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],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,16,0,0,16,4,0,0,0,0,0,1,0,0],[14,14,0,0,0,0,14,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.269D4 in GAP, Magma, Sage, TeX
C_4^2._{269}D_4 % in TeX
G:=Group("C4^2.269D4"); // GroupNames label
G:=SmallGroup(128,1943);
// by ID
G=gap.SmallGroup(128,1943);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=b^2*c^3>;
// generators/relations