p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23⋊3D8, C24.119D4, C4.12+ (1+4), C8⋊7D4⋊3C2, C22⋊D8⋊4C2, (C2×D8)⋊2C22, C2.D8⋊3C22, (C22×C8)⋊9C22, C2.14(C22×D8), C22.23(C2×D8), D4⋊C4⋊1C22, C4⋊C4.125C23, C4⋊D4⋊56C22, C22⋊C8⋊53C22, (C2×C8).150C23, (C2×C4).384C24, (C22×C4).482D4, C22.D8⋊4C2, C23.399(C2×D4), (C2×D4).137C23, (C22×D4)⋊23C22, C2.65(C23⋊3D4), (C23×C4).564C22, C22.644(C22×D4), C2.47(D8⋊C22), (C22×C4).1062C23, (C2×C4⋊D4)⋊49C2, (C2×C22⋊C8)⋊22C2, (C2×C4⋊C4)⋊50C22, (C2×C4).525(C2×D4), SmallGroup(128,1918)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 660 in 260 conjugacy classes, 94 normal (14 characteristic)
C1, C2 [×3], C2 [×10], C4 [×2], C4 [×7], C22, C22 [×6], C22 [×30], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×17], D4 [×28], C23, C23 [×6], C23 [×18], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], D8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C2×D4 [×4], C2×D4 [×22], C24, C24 [×2], C22⋊C8 [×4], D4⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4⋊D4 [×8], C4⋊D4 [×4], C22×C8 [×2], C2×D8 [×4], C23×C4, C22×D4 [×2], C22×D4 [×2], C2×C22⋊C8, C22⋊D8 [×4], C8⋊7D4 [×4], C22.D8 [×4], C2×C4⋊D4 [×2], C23⋊3D8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, 2+ (1+4) [×2], C23⋊3D4, C22×D8, D8⋊C22, C23⋊3D8
Generators and relations
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, eae=ac=ca, ad=da, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
►On 32 points
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(1 5)(2 26)(3 7)(4 28)(6 30)(8 32)(9 23)(10 14)(11 17)(12 16)(13 19)(15 21)(18 22)(20 24)(25 29)(27 31)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (1,5)(2,26)(3,7)(4,28)(6,30)(8,32)(9,23)(10,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,24)(25,29)(27,31), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,17)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29), (1,5)(2,26)(3,7)(4,28)(6,30)(8,32)(9,23)(10,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,24)(25,29)(27,31), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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([(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,17),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29)], [(1,5),(2,26),(3,7),(4,28),(6,30),(8,32),(9,23),(10,14),(11,17),(12,16),(13,19),(15,21),(18,22),(20,24),(25,29),(27,31)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18)], [(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)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 3 | 14 | 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))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,1,0,0,0],[3,3,0,0,0,0,3,14,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | 2+ (1+4) | D8⋊C22 |
| kernel | C23⋊3D8 | C2×C22⋊C8 | C22⋊D8 | C8⋊7D4 | C22.D8 | C2×C4⋊D4 | C22×C4 | C24 | C23 | C4 | C2 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 3 | 1 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2^3\rtimes_3D_8
% in TeX
G:=Group("C2^3:3D8"); // GroupNames label
G:=SmallGroup(128,1918);
// by ID
G=gap.SmallGroup(128,1918);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,e*a*e=a*c=c*a,a*d=d*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations