p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.269D4, C42.730C23, C4.1032+ (1+4), C4⋊C8⋊67C22, (C4×C8)⋊16C22, Q8⋊D4⋊31C2, D4.2D4⋊2C2, C22⋊D8.2C2, C8.12D4⋊3C2, (C4×D4)⋊12C22, Q8.D4⋊2C2, C22⋊Q16⋊6C2, (C2×Q16)⋊5C22, (C4×Q8)⋊12C22, C22⋊SD16⋊31C2, C4⋊C4.150C23, (C2×C8).327C23, (C2×C4).409C24, Q8⋊C4⋊5C22, (C2×D8).24C22, (C22×C4).172D4, C23.692(C2×D4), (C2×SD16)⋊43C22, (C2×D4).158C23, C22.33(C4○D8), C4.4D4⋊60C22, (C2×Q8).146C23, C42.C2⋊37C22, C42.12C4⋊35C2, C4⋊D4.189C22, C22⋊C8.179C22, (C2×C42).876C22, C22.669(C22×D4), C22⋊Q8.194C22, D4⋊C4.107C22, C2.54(D8⋊C22), C23.36C23⋊9C2, C42.78C22⋊7C2, (C22×C4).1080C23, (C22×D4).388C22, (C22×Q8).321C22, C2.80(C22.29C24), C2.43(C2×C4○D8), (C2×C4).539(C2×D4), (C2×C4.4D4)⋊43C2, SmallGroup(128,1943)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 452 in 205 conjugacy classes, 86 normal (42 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×15], C8 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×10], Q8 [×8], C23, C23 [×9], C42 [×4], C42, C22⋊C4 [×13], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C2×Q8 [×2], C2×Q8 [×3], C24, C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4 [×4], C4.4D4 [×2], C42.C2, C42⋊2C2, C2×D8 [×2], C2×SD16 [×4], C2×Q16 [×2], C22×D4, C22×Q8, C42.12C4, C22⋊D8, Q8⋊D4, C22⋊SD16, C22⋊Q16, D4.2D4 [×2], Q8.D4 [×2], C42.78C22 [×2], C8.12D4 [×2], C2×C4.4D4, C23.36C23, C42.269D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2+ (1+4) [×2], C22.29C24, C2×C4○D8, D8⋊C22, C42.269D4
Generators and relations
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 >
►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)])
Matrix representation ►G ⊆ 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] >;
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 | C4○D8 | 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 | C2×C4.4D4 | C23.36C23 | C42 | C22×C4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
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