p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.677C23, C4⋊C8⋊83C22, (C4×C8)⋊52C22, (C2×C4)⋊12M4(2), (C23×C4).42C4, (C2×C42).57C4, C8⋊C4⋊54C22, (C4×M4(2))⋊29C2, C42.334(C2×C4), (C2×C4).639C24, C24.130(C2×C4), (C2×C8).397C23, C4.54(C2×M4(2)), C4○2(C4⋊M4(2)), C4⋊M4(2)⋊41C2, C4○2(C24.4C4), C4○2(C42.6C4), C42.6C4⋊58C2, (C22×C42).34C2, C42.12C4⋊45C2, C2.9(C22×M4(2)), C4.65(C42⋊C2), C22⋊C8.227C22, C42○(C4⋊M4(2)), C24.4C4.25C2, C42○(C24.4C4), C42○(C42.6C4), C22.167(C23×C4), C23.225(C22×C4), (C23×C4).698C22, C22.28(C2×M4(2)), (C2×C42).1106C22, (C22×C4).1272C23, C22.37(C42⋊C2), (C2×M4(2)).341C22, C4.290(C2×C4○D4), (C2×C4).677(C4○D4), (C22×C4).498(C2×C4), (C2×C4).498(C22×C4), C2.39(C2×C42⋊C2), SmallGroup(128,1652)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 332 in 242 conjugacy classes, 152 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×14], C8 [×8], C2×C4 [×2], C2×C4 [×22], C2×C4 [×30], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×12], C2×C8 [×8], M4(2) [×8], C22×C4 [×2], C22×C4 [×16], C22×C4 [×12], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×2], C2×C42 [×10], C2×M4(2) [×4], C23×C4, C23×C4 [×2], C4×M4(2) [×2], C24.4C4 [×2], C4⋊M4(2) [×2], C42.12C4 [×4], C42.6C4 [×4], C22×C42, C42.677C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×8], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C2×M4(2) [×12], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C22×M4(2) [×2], C42.677C23
Generators and relations
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2, ab=ba, cac-1=a-1b2, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, de=ed >
►On 32 points
(1 14 31 23)(2 20 32 11)(3 16 25 17)(4 22 26 13)(5 10 27 19)(6 24 28 15)(7 12 29 21)(8 18 30 9)
(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 19 31 10)(2 20 32 11)(3 21 25 12)(4 22 26 13)(5 23 27 14)(6 24 28 15)(7 17 29 16)(8 18 30 9)
(1 19 31 10)(2 11 32 20)(3 21 25 12)(4 13 26 22)(5 23 27 14)(6 15 28 24)(7 17 29 16)(8 9 30 18)
G:=sub<Sym(32)| (1,14,31,23)(2,20,32,11)(3,16,25,17)(4,22,26,13)(5,10,27,19)(6,24,28,15)(7,12,29,21)(8,18,30,9), (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), (1,19,31,10)(2,20,32,11)(3,21,25,12)(4,22,26,13)(5,23,27,14)(6,24,28,15)(7,17,29,16)(8,18,30,9), (1,19,31,10)(2,11,32,20)(3,21,25,12)(4,13,26,22)(5,23,27,14)(6,15,28,24)(7,17,29,16)(8,9,30,18)>;
G:=Group( (1,14,31,23)(2,20,32,11)(3,16,25,17)(4,22,26,13)(5,10,27,19)(6,24,28,15)(7,12,29,21)(8,18,30,9), (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), (1,19,31,10)(2,20,32,11)(3,21,25,12)(4,22,26,13)(5,23,27,14)(6,24,28,15)(7,17,29,16)(8,18,30,9), (1,19,31,10)(2,11,32,20)(3,21,25,12)(4,13,26,22)(5,23,27,14)(6,15,28,24)(7,17,29,16)(8,9,30,18) );
G=PermutationGroup([(1,14,31,23),(2,20,32,11),(3,16,25,17),(4,22,26,13),(5,10,27,19),(6,24,28,15),(7,12,29,21),(8,18,30,9)], [(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)], [(1,19,31,10),(2,20,32,11),(3,21,25,12),(4,22,26,13),(5,23,27,14),(6,24,28,15),(7,17,29,16),(8,18,30,9)], [(1,19,31,10),(2,11,32,20),(3,21,25,12),(4,13,26,22),(5,23,27,14),(6,15,28,24),(7,17,29,16),(8,9,30,18)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,4],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,13,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | ··· | 4AD | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | M4(2) | C4○D4 |
| kernel | C42.677C23 | C4×M4(2) | C24.4C4 | C4⋊M4(2) | C42.12C4 | C42.6C4 | C22×C42 | C2×C42 | C23×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 12 | 4 | 16 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{677}C_2^3 % in TeX
G:=Group("C4^2.677C2^3"); // GroupNames label
G:=SmallGroup(128,1652);
// by ID
G=gap.SmallGroup(128,1652);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,2019,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=b,d^2=e^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,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,d*e=e*d>;
// generators/relations