p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.115D4, C4.Q8⋊64C22, C2.D8⋊55C22, C4⋊C4.390C23, (C2×C8).143C23, (C2×C4).289C24, (C2×D4).78C23, C4○(C22.D8), (C22×C4).440D4, C23.241(C2×D4), (C2×Q8).66C23, D4⋊C4⋊78C22, Q8⋊C4⋊82C22, C4○(C23.19D4), C4○(C23.46D4), C4○(C23.20D4), C4○(C23.47D4), C22.28(C4○D8), C4○(C23.48D4), C23.25D4⋊6C2, C22.D8⋊37C2, C23.19D4⋊51C2, C23.48D4⋊37C2, C23.24D4⋊16C2, C23.47D4⋊35C2, C4⋊D4.154C22, C23.20D4⋊52C2, C23.46D4⋊35C2, C22⋊C8.172C22, (C22×C8).183C22, (C23×C4).559C22, C22.549(C22×D4), C22⋊Q8.159C22, C2.21(D8⋊C22), C22.19C24.17C2, (C22×C4).1547C23, C42⋊C2.122C22, C4.144(C22.D4), C22.17(C22.D4), C2.22(C2×C4○D8), C4.99(C2×C4○D4), (C2×C22⋊C8)⋊27C2, (C2×C4).485(C2×D4), (C2×C42⋊C2)⋊45C2, (C2×C4).847(C4○D4), (C2×C4⋊C4).926C22, (C2×C4)○(C22.D8), (C2×C4○D4).136C22, (C2×C4)○(C23.46D4), (C2×C4)○(C23.19D4), (C2×C4)○(C23.47D4), (C2×C4)○(C23.20D4), C2.54(C2×C22.D4), SmallGroup(128,1823)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 388 in 210 conjugacy classes, 94 normal (42 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×14], C8 [×4], C2×C4 [×4], C2×C4 [×4], C2×C4 [×23], D4 [×8], Q8 [×2], C23 [×3], C23 [×6], C42 [×5], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×4], C22×C4 [×6], C22×C4 [×7], C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], C24, C22⋊C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2.D8 [×4], C2×C42, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2 [×4], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22×C8 [×2], C23×C4, C2×C4○D4, C2×C22⋊C8, C23.24D4 [×2], C23.25D4 [×2], C22.D8, C23.46D4, C23.19D4 [×2], C23.47D4, C23.48D4, C23.20D4 [×2], C2×C42⋊C2, C22.19C24, C24.115D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C4○D8 [×2], C22×D4, C2×C4○D4 [×2], C2×C22.D4, C2×C4○D8, D8⋊C22, C24.115D4
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, fbf=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >
►On 32 points
(2 31)(4 25)(6 27)(8 29)(9 22)(11 24)(13 18)(15 20)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 22)(2 16)(3 20)(4 14)(5 18)(6 12)(7 24)(8 10)(9 30)(11 28)(13 26)(15 32)(17 27)(19 25)(21 31)(23 29)
G:=sub<Sym(32)| (2,31)(4,25)(6,27)(8,29)(9,22)(11,24)(13,18)(15,20), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,22)(2,16)(3,20)(4,14)(5,18)(6,12)(7,24)(8,10)(9,30)(11,28)(13,26)(15,32)(17,27)(19,25)(21,31)(23,29)>;
G:=Group( (2,31)(4,25)(6,27)(8,29)(9,22)(11,24)(13,18)(15,20), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,22)(2,16)(3,20)(4,14)(5,18)(6,12)(7,24)(8,10)(9,30)(11,28)(13,26)(15,32)(17,27)(19,25)(21,31)(23,29) );
G=PermutationGroup([(2,31),(4,25),(6,27),(8,29),(9,22),(11,24),(13,18),(15,20)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,22),(2,16),(3,20),(4,14),(5,18),(6,12),(7,24),(8,10),(9,30),(11,28),(13,26),(15,32),(17,27),(19,25),(21,31),(23,29)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 15 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 13 | 9 |
| 0 | 0 | 4 | 4 |
| 0 | 8 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,0,16],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,8,0,0,0,0,13,4,0,0,9,4],[0,15,0,0,8,0,0,0,0,0,16,0,0,0,15,1] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4Q | 4R | 4S | 4T | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C4○D8 | D8⋊C22 |
| kernel | C24.115D4 | C2×C22⋊C8 | C23.24D4 | C23.25D4 | C22.D8 | C23.46D4 | C23.19D4 | C23.47D4 | C23.48D4 | C23.20D4 | C2×C42⋊C2 | C22.19C24 | C22×C4 | C24 | C2×C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 3 | 1 | 8 | 8 | 2 |
In GAP, Magma, Sage, TeX
C_2^4._{115}D_4 % in TeX
G:=Group("C2^4.115D4"); // GroupNames label
G:=SmallGroup(128,1823);
// by ID
G=gap.SmallGroup(128,1823);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*d*e^3>;
// generators/relations