p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.240D4, C42.707C23, (C4×C8)⋊64C22, C4⋊Q8⋊63C22, C4.4D8⋊40C2, C4⋊C4.91C23, C8⋊C4⋊65C22, C4.20(C8⋊C22), (C4×M4(2))⋊40C2, (C2×C4).336C24, (C2×C8).457C23, C23.678(C2×D4), (C22×C4).460D4, D4⋊C4⋊53C22, C4.77(C4.4D4), (C2×D4).104C23, C42.C2⋊35C22, C4⋊1D4.147C22, C23.37D4⋊37C2, (C2×C42).847C22, C22.596(C22×D4), (C22×C4).1034C23, C22.31(C4.4D4), (C22×D4).369C22, C42⋊C2.141C22, C42.29C22⋊18C2, C23.37C23⋊10C2, (C2×M4(2)).373C22, C4.45(C2×C4○D4), (C2×C4).514(C2×D4), C2.39(C2×C8⋊C22), (C2×C4⋊1D4).24C2, C2.47(C2×C4.4D4), (C2×C4).300(C4○D4), SmallGroup(128,1870)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 580 in 242 conjugacy classes, 96 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×8], D4 [×28], Q8 [×4], C23, C23 [×16], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×26], C2×Q8 [×2], C24 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×16], C2×C42, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊1D4 [×4], C4⋊1D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×D4 [×2], C22×D4 [×2], C4×M4(2), C23.37D4 [×4], C4.4D8 [×4], C42.29C22 [×4], C2×C4⋊1D4, C23.37C23, C42.240D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C8⋊C22 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, C2×C8⋊C22 [×2], C42.240D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2c3 >
►On 32 points
(1 18 25 11)(2 19 26 12)(3 20 27 13)(4 21 28 14)(5 22 29 15)(6 23 30 16)(7 24 31 9)(8 17 32 10)
(1 31 5 27)(2 28 6 32)(3 25 7 29)(4 30 8 26)(9 22 13 18)(10 19 14 23)(11 24 15 20)(12 21 16 17)
(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 23 29 12)(2 11 30 22)(3 21 31 10)(4 9 32 20)(5 19 25 16)(6 15 26 18)(7 17 27 14)(8 13 28 24)
G:=sub<Sym(32)| (1,18,25,11)(2,19,26,12)(3,20,27,13)(4,21,28,14)(5,22,29,15)(6,23,30,16)(7,24,31,9)(8,17,32,10), (1,31,5,27)(2,28,6,32)(3,25,7,29)(4,30,8,26)(9,22,13,18)(10,19,14,23)(11,24,15,20)(12,21,16,17), (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), (1,23,29,12)(2,11,30,22)(3,21,31,10)(4,9,32,20)(5,19,25,16)(6,15,26,18)(7,17,27,14)(8,13,28,24)>;
G:=Group( (1,18,25,11)(2,19,26,12)(3,20,27,13)(4,21,28,14)(5,22,29,15)(6,23,30,16)(7,24,31,9)(8,17,32,10), (1,31,5,27)(2,28,6,32)(3,25,7,29)(4,30,8,26)(9,22,13,18)(10,19,14,23)(11,24,15,20)(12,21,16,17), (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), (1,23,29,12)(2,11,30,22)(3,21,31,10)(4,9,32,20)(5,19,25,16)(6,15,26,18)(7,17,27,14)(8,13,28,24) );
G=PermutationGroup([(1,18,25,11),(2,19,26,12),(3,20,27,13),(4,21,28,14),(5,22,29,15),(6,23,30,16),(7,24,31,9),(8,17,32,10)], [(1,31,5,27),(2,28,6,32),(3,25,7,29),(4,30,8,26),(9,22,13,18),(10,19,14,23),(11,24,15,20),(12,21,16,17)], [(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)], [(1,23,29,12),(2,11,30,22),(3,21,31,10),(4,9,32,20),(5,19,25,16),(6,15,26,18),(7,17,27,14),(8,13,28,24)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 9 | 0 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,13,0,0,0,0,9,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,16,0,0,0],[13,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 |
| kernel | C42.240D4 | C4×M4(2) | C23.37D4 | C4.4D8 | C42.29C22 | C2×C4⋊1D4 | C23.37C23 | C42 | C22×C4 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{240}D_4 % in TeX
G:=Group("C4^2.240D4"); // GroupNames label
G:=SmallGroup(128,1870);
// by ID
G=gap.SmallGroup(128,1870);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,100,1018,521,2804,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations