p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.262C23, C4⋊C8⋊84C22, (C4×C8)⋊54C22, C24.78(C2×C4), C8⋊C4⋊56C22, C42.206(C2×C4), (C2×C4).643C24, (C2×C8).475C23, C8○2M4(2)⋊28C2, C42⋊C2.30C4, C42.6C4⋊45C2, C22.15(C8○D4), C42.12C4⋊47C2, C4.30(C42⋊C2), C22⋊C8.229C22, C2.12(Q8○M4(2)), C24.4C4.24C2, C23.226(C22×C4), (C23×C4).523C22, (C2×C42).756C22, C22.171(C23×C4), (C22×C8).431C22, C42.6C22⋊29C2, C42.7C22⋊21C2, (C22×C4).1273C23, C42⋊C2.349C22, C22.13(C42⋊C2), (C2×M4(2)).345C22, (C2×C4⋊C4).69C4, C2.12(C2×C8○D4), C4⋊C4.219(C2×C4), C4.294(C2×C4○D4), (C2×C22⋊C8).47C2, C22⋊C4.89(C2×C4), (C2×C22⋊C4).46C4, (C2×C4).828(C4○D4), C22⋊C8○(C42⋊C2), (C22×C4).337(C2×C4), (C2×C4).259(C22×C4), C2.43(C2×C42⋊C2), (C2×C42⋊C2).58C2, SmallGroup(128,1656)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 284 in 201 conjugacy classes, 134 normal (36 characteristic)
C1, C2 [×3], C2 [×5], C4 [×4], C4 [×10], C22, C22 [×4], C22 [×11], C8 [×8], C2×C4 [×4], C2×C4 [×12], C2×C4 [×16], C23 [×3], C23 [×5], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×4], C22×C4 [×4], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×2], C22⋊C8 [×6], C4⋊C8 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×4], C22×C8 [×2], C2×M4(2) [×2], C23×C4, C8○2M4(2) [×2], C2×C22⋊C8, C24.4C4, C42.6C22 [×2], C42.12C4 [×2], C42.6C4 [×2], C42.7C22 [×4], C2×C42⋊C2, C42.262C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C2×C8○D4, Q8○M4(2), C42.262C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, de=ed >
►On 32 points
(1 19 31 15)(2 16 32 20)(3 21 25 9)(4 10 26 22)(5 23 27 11)(6 12 28 24)(7 17 29 13)(8 14 30 18)
(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)
(2 32)(4 26)(6 28)(8 30)(10 22)(12 24)(14 18)(16 20)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
G:=sub<Sym(32)| (1,19,31,15)(2,16,32,20)(3,21,25,9)(4,10,26,22)(5,23,27,11)(6,12,28,24)(7,17,29,13)(8,14,30,18), (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), (2,32)(4,26)(6,28)(8,30)(10,22)(12,24)(14,18)(16,20), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)>;
G:=Group( (1,19,31,15)(2,16,32,20)(3,21,25,9)(4,10,26,22)(5,23,27,11)(6,12,28,24)(7,17,29,13)(8,14,30,18), (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), (2,32)(4,26)(6,28)(8,30)(10,22)(12,24)(14,18)(16,20), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24) );
G=PermutationGroup([(1,19,31,15),(2,16,32,20),(3,21,25,9),(4,10,26,22),(5,23,27,11),(6,12,28,24),(7,17,29,13),(8,14,30,18)], [(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)], [(2,32),(4,26),(6,28),(8,30),(10,22),(12,24),(14,18),(16,20)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 9 | 13 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 7 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,4,9,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,2,7,0,0,2,15],[1,0,0,0,0,1,0,0,0,0,1,15,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4U | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4○D4 | C8○D4 | Q8○M4(2) |
| kernel | C42.262C23 | C8○2M4(2) | C2×C22⋊C8 | C24.4C4 | C42.6C22 | C42.12C4 | C42.6C4 | C42.7C22 | C2×C42⋊C2 | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊C2 | C2×C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 4 | 4 | 8 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{262}C_2^3 % in TeX
G:=Group("C4^2.262C2^3"); // GroupNames label
G:=SmallGroup(128,1656);
// by ID
G=gap.SmallGroup(128,1656);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,521,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,d*e=e*d>;
// generators/relations