p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.283C23, M4(2).32C23, C8○D8⋊5C2, C8○(C8⋊C22), C8⋊C22⋊7C4, D8.6(C2×C4), C8.59(C2×D4), SD16.(C2×C4), C8○(C8.26D4), C8.26D4⋊6C2, (C4×C8)⋊22C22, C4.135(C4×D4), (C2×C8).395D4, C8○(C8.C22), C8.C22⋊7C4, C8.4(C22×C4), C4≀C2⋊18C22, Q16.6(C2×C4), C8○D4⋊19C22, C4.32(C23×C4), C22.48(C4×D4), C8⋊C4⋊41C22, C8○(D8⋊C22), (C2×C4).212C24, (C2×C8).613C23, C4○D4.24C23, C4○D8.24C22, D4.14(C22×C4), C4.203(C22×D4), Q8.14(C22×C4), C8○(M4(2).C4), C8○2M4(2)⋊10C2, C8.C4⋊12C22, C8○(C42⋊C22), D8⋊C22.9C2, M4(2).13(C2×C4), M4(2).C4⋊15C2, C42⋊C22⋊21C2, C23.110(C4○D4), (C22×C4).931C23, (C22×C8).442C22, C42⋊C2.301C22, (C2×M4(2)).359C22, C2.72(C2×C4×D4), (C2×C8○D4)⋊27C2, C4○D4.23(C2×C4), C22.3(C2×C4○D4), (C2×D4).179(C2×C4), (C2×C4).1413(C2×D4), (C2×C4).72(C22×C4), (C2×Q8).162(C2×C4), (C2×C4).269(C4○D4), (C2×C4○D4).295C22, SmallGroup(128,1687)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 340 in 228 conjugacy classes, 138 normal (20 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×9], C8 [×2], C8 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C42 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×6], C2×C8 [×10], M4(2) [×8], M4(2) [×10], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], C4≀C2 [×8], C8.C4 [×4], C42⋊C2, C22×C8, C22×C8 [×2], C2×M4(2), C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×8], C8○D4 [×4], C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C8○2M4(2), C42⋊C22 [×2], M4(2).C4, C8○D8 [×4], C8.26D4 [×4], C2×C8○D4 [×2], D8⋊C22, C42.283C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C42.283C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=b, e2=a2, ab=ba, cac-1=a-1b-1, ad=da, eae-1=ab2, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2b-1c, de=ed >
►On 32 points
(1 14 21 29)(2 15 22 30)(3 16 23 31)(4 9 24 32)(5 10 17 25)(6 11 18 26)(7 12 19 27)(8 13 20 28)
(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 13 5 9)(2 14 6 10)(3 15 7 11)(4 16 8 12)(17 28 21 32)(18 29 22 25)(19 30 23 26)(20 31 24 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 16 21 31)(2 9 22 32)(3 10 23 25)(4 11 24 26)(5 12 17 27)(6 13 18 28)(7 14 19 29)(8 15 20 30)
G:=sub<Sym(32)| (1,14,21,29)(2,15,22,30)(3,16,23,31)(4,9,24,32)(5,10,17,25)(6,11,18,26)(7,12,19,27)(8,13,20,28), (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), (1,13,5,9)(2,14,6,10)(3,15,7,11)(4,16,8,12)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27), (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), (1,16,21,31)(2,9,22,32)(3,10,23,25)(4,11,24,26)(5,12,17,27)(6,13,18,28)(7,14,19,29)(8,15,20,30)>;
G:=Group( (1,14,21,29)(2,15,22,30)(3,16,23,31)(4,9,24,32)(5,10,17,25)(6,11,18,26)(7,12,19,27)(8,13,20,28), (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), (1,13,5,9)(2,14,6,10)(3,15,7,11)(4,16,8,12)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27), (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), (1,16,21,31)(2,9,22,32)(3,10,23,25)(4,11,24,26)(5,12,17,27)(6,13,18,28)(7,14,19,29)(8,15,20,30) );
G=PermutationGroup([(1,14,21,29),(2,15,22,30),(3,16,23,31),(4,9,24,32),(5,10,17,25),(6,11,18,26),(7,12,19,27),(8,13,20,28)], [(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)], [(1,13,5,9),(2,14,6,10),(3,15,7,11),(4,16,8,12),(17,28,21,32),(18,29,22,25),(19,30,23,26),(20,31,24,27)], [(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)], [(1,16,21,31),(2,9,22,32),(3,10,23,25),(4,11,24,26),(5,12,17,27),(6,13,18,28),(7,14,19,29),(8,15,20,30)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 9 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 |
| 0 | 13 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
G:=sub<GL(4,GF(17))| [0,0,0,9,0,0,9,0,8,0,0,0,0,9,0,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,0,0,4,0,0,13,0,0,13,0,0,4,0,0,0],[15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,0,0,2,0,0,2,0,2,0,0,0,0,15,0,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | ··· | 8V |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | C4○D4 | C4○D4 | C42.283C23 |
| kernel | C42.283C23 | C8○2M4(2) | C42⋊C22 | M4(2).C4 | C8○D8 | C8.26D4 | C2×C8○D4 | D8⋊C22 | C8⋊C22 | C8.C22 | C2×C8 | C2×C4 | C23 | C1 |
| # reps | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 1 | 8 | 8 | 4 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{283}C_2^3 % in TeX
G:=Group("C4^2.283C2^3"); // GroupNames label
G:=SmallGroup(128,1687);
// by ID
G=gap.SmallGroup(128,1687);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,2804,1411,172,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b^2,d^2=b,e^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,a*d=d*a,e*a*e^-1=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*b^-1*c,d*e=e*d>;
// generators/relations