p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.263D4, C42.395C23, C4.972+ 1+4, (C2×C4)⋊4D8, C4⋊D8⋊7C2, C8⋊4D4⋊8C2, (C4×C8)⋊9C22, C4.72(C2×D8), C22⋊D8⋊6C2, C4⋊C8⋊60C22, (C2×D8)⋊5C22, C4⋊Q8⋊69C22, C4.4D8⋊11C2, (C4×D4)⋊11C22, C2.15(C22×D8), C22.24(C2×D8), D4⋊C4⋊3C22, C4⋊C4.144C23, C4⋊1D4⋊40C22, C4.25(C8⋊C22), (C2×C4).403C24, (C2×C8).158C23, (C22×C4).493D4, C23.688(C2×D4), (C2×D4).153C23, C42.12C4⋊26C2, C4⋊D4.186C22, C22⋊C8.177C22, (C2×C42).870C22, C22.663(C22×D4), (C22×C4).1074C23, C22.26C24⋊16C2, (C22×D4).386C22, C2.74(C22.29C24), (C2×C4⋊1D4)⋊20C2, (C2×C4).864(C2×D4), C2.53(C2×C8⋊C22), SmallGroup(128,1937)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.263D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=b2c3 >
Subgroups: 700 in 266 conjugacy classes, 96 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×D8, C22×D4, C22×D4, C2×C4○D4, C42.12C4, C22⋊D8, C4⋊D8, C4.4D8, C8⋊4D4, C2×C4⋊1D4, C22.26C24, C42.263D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C8⋊C22, C22×D4, 2+ 1+4, C22.29C24, C22×D8, C2×C8⋊C22, C42.263D4
►On 32 points
(1 12 26 23)(2 13 27 24)(3 14 28 17)(4 15 29 18)(5 16 30 19)(6 9 31 20)(7 10 32 21)(8 11 25 22)
(1 7 5 3)(2 29 6 25)(4 31 8 27)(9 22 13 18)(10 16 14 12)(11 24 15 20)(17 23 21 19)(26 32 30 28)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(8 16)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)
G:=sub<Sym(32)| (1,12,26,23)(2,13,27,24)(3,14,28,17)(4,15,29,18)(5,16,30,19)(6,9,31,20)(7,10,32,21)(8,11,25,22), (1,7,5,3)(2,29,6,25)(4,31,8,27)(9,22,13,18)(10,16,14,12)(11,24,15,20)(17,23,21,19)(26,32,30,28), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(8,16)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)>;
G:=Group( (1,12,26,23)(2,13,27,24)(3,14,28,17)(4,15,29,18)(5,16,30,19)(6,9,31,20)(7,10,32,21)(8,11,25,22), (1,7,5,3)(2,29,6,25)(4,31,8,27)(9,22,13,18)(10,16,14,12)(11,24,15,20)(17,23,21,19)(26,32,30,28), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(8,16)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28) );
G=PermutationGroup([[(1,12,26,23),(2,13,27,24),(3,14,28,17),(4,15,29,18),(5,16,30,19),(6,9,31,20),(7,10,32,21),(8,11,25,22)], [(1,7,5,3),(2,29,6,25),(4,31,8,27),(9,22,13,18),(10,16,14,12),(11,24,15,20),(17,23,21,19),(26,32,30,28)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(8,16),(17,27),(18,26),(19,25),(20,32),(21,31),(22,30),(23,29),(24,28)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | C8⋊C22 | 2+ 1+4 |
| kernel | C42.263D4 | C42.12C4 | C22⋊D8 | C4⋊D8 | C4.4D8 | C8⋊4D4 | C2×C4⋊1D4 | C22.26C24 | C42 | C22×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.263D4 ►in GL6(𝔽17)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 2 | 4 | 16 | 2 |
| 0 | 0 | 0 | 2 | 16 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 | 16 | 0 |
| 0 | 0 | 15 | 15 | 0 | 16 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 1 | 15 |
| 0 | 0 | 0 | 1 | 16 | 1 |
| 0 | 0 | 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 1 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,16,1,2,0,0,0,15,1,4,2,0,0,0,0,16,16,0,0,0,0,2,1],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,1,0,15,15,0,0,0,1,0,15,0,0,0,0,16,0,0,0,0,0,0,16],[6,14,0,0,0,0,6,0,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,1,16,16,0,0,0,15,1,2,1],[0,3,0,0,0,0,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,16,1,1,0,0,0,0,16,0,1] >;
C42.263D4 in GAP, Magma, Sage, TeX
C_4^2._{263}D_4 % in TeX
G:=Group("C4^2.263D4"); // GroupNames label
G:=SmallGroup(128,1937);
// by ID
G=gap.SmallGroup(128,1937);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,675,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=b^2*c^3>;
// generators/relations