p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.242D4, C42.368C23, (C4×C8)⋊65C22, C4⋊Q8⋊64C22, C4⋊C4.93C23, C8⋊C4⋊66C22, (C4×M4(2))⋊41C2, (C2×C4).338C24, (C2×C8).459C23, C23.680(C2×D4), (C22×C4).462D4, (C2×Q8).93C23, Q8⋊C4⋊55C22, C4.79(C4.4D4), (C2×D4).105C23, C42.C2⋊36C22, C23.38D4⋊37C2, (C2×C42).849C22, C22.598(C22×D4), D4⋊C4.133C22, C2.36(D8⋊C22), (C22×C4).1036C23, C4.4D4.136C22, C23.37D4.11C2, C22.33(C4.4D4), (C22×D4).370C22, (C22×Q8).303C22, C23.37C23⋊11C2, C42.28C22⋊31C2, C42⋊C2.143C22, C42.78C22⋊27C2, (C2×M4(2)).375C22, C4.47(C2×C4○D4), (C2×C4).516(C2×D4), C2.49(C2×C4.4D4), (C2×C4).302(C4○D4), (C2×C4.4D4).40C2, SmallGroup(128,1872)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.242D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=a2c3 >
Subgroups: 404 in 200 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C4.4D4, C4.4D4, C42.C2, C4⋊Q8, C2×M4(2), C22×D4, C22×Q8, C4×M4(2), C23.37D4, C23.38D4, C42.78C22, C42.28C22, C2×C4.4D4, C23.37C23, C42.242D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, C2×C4.4D4, D8⋊C22, C42.242D4
►On 32 points
(1 21 29 15)(2 22 30 16)(3 23 31 9)(4 24 32 10)(5 17 25 11)(6 18 26 12)(7 19 27 13)(8 20 28 14)
(1 27 5 31)(2 32 6 28)(3 29 7 25)(4 26 8 30)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 24)
(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 32 25 8)(2 7 26 31)(3 30 27 6)(4 5 28 29)(9 12 19 22)(10 21 20 11)(13 16 23 18)(14 17 24 15)
G:=sub<Sym(32)| (1,21,29,15)(2,22,30,16)(3,23,31,9)(4,24,32,10)(5,17,25,11)(6,18,26,12)(7,19,27,13)(8,20,28,14), (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,21,13,17)(10,18,14,22)(11,23,15,19)(12,20,16,24), (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), (1,32,25,8)(2,7,26,31)(3,30,27,6)(4,5,28,29)(9,12,19,22)(10,21,20,11)(13,16,23,18)(14,17,24,15)>;
G:=Group( (1,21,29,15)(2,22,30,16)(3,23,31,9)(4,24,32,10)(5,17,25,11)(6,18,26,12)(7,19,27,13)(8,20,28,14), (1,27,5,31)(2,32,6,28)(3,29,7,25)(4,26,8,30)(9,21,13,17)(10,18,14,22)(11,23,15,19)(12,20,16,24), (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), (1,32,25,8)(2,7,26,31)(3,30,27,6)(4,5,28,29)(9,12,19,22)(10,21,20,11)(13,16,23,18)(14,17,24,15) );
G=PermutationGroup([[(1,21,29,15),(2,22,30,16),(3,23,31,9),(4,24,32,10),(5,17,25,11),(6,18,26,12),(7,19,27,13),(8,20,28,14)], [(1,27,5,31),(2,32,6,28),(3,29,7,25),(4,26,8,30),(9,21,13,17),(10,18,14,22),(11,23,15,19),(12,20,16,24)], [(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)], [(1,32,25,8),(2,7,26,31),(3,30,27,6),(4,5,28,29),(9,12,19,22),(10,21,20,11),(13,16,23,18),(14,17,24,15)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 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 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D8⋊C22 |
| kernel | C42.242D4 | C4×M4(2) | C23.37D4 | C23.38D4 | C42.78C22 | C42.28C22 | C2×C4.4D4 | C23.37C23 | C42 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 8 | 4 |
Matrix representation of C42.242D4 ►in GL6(𝔽17)
| 16 | 8 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 1 |
| 0 | 0 | 0 | 13 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 4 | 13 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 16 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 9 | 0 |
| 0 | 0 | 1 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 13 | 4 | 16 | 0 |
G:=sub<GL(6,GF(17))| [16,4,0,0,0,0,8,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,4,0,0,0,2,1,13,13,0,0,0,0,0,16,0,0,0,0,1,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,4,0,0,0,0,0,13,0,0,0,8,4,1,1,0,0,0,13,0,0],[13,16,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,13,0,0,0,0,0,4,0,0,9,13,16,16,0,0,0,13,0,0] >;
C42.242D4 in GAP, Magma, Sage, TeX
C_4^2._{242}D_4 % in TeX
G:=Group("C4^2.242D4"); // GroupNames label
G:=SmallGroup(128,1872);
// by ID
G=gap.SmallGroup(128,1872);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,680,758,100,1018,521,2804,172,4037,124]);
// 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*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c^3>;
// generators/relations