p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.219D4, C42.333C23, C4⋊C8⋊9C22, (C2×D4)⋊22Q8, D4.3(C2×Q8), D4⋊2Q8⋊1C2, D4.Q8⋊12C2, C4⋊Q8⋊56C22, C4.Q8⋊6C22, D4⋊Q8⋊18C2, C4⋊C4.40C23, (C2×C8).24C23, C2.D8⋊17C22, C4.28(C22×Q8), C4⋊M4(2)⋊7C2, (C2×C4).275C24, C23.657(C2×D4), (C22×C4).799D4, C4.101(C8⋊C22), (C2×D4).394C23, (C4×D4).316C22, C4.106(C22⋊Q8), C42.C2⋊30C22, M4(2)⋊C4⋊16C2, D4⋊C4.23C22, (C2×C42).821C22, (C22×C4).994C23, C23.37D4.2C2, C22.535(C22×D4), C22.45(C22⋊Q8), C2.18(D8⋊C22), C23.37C23⋊3C2, (C22×D4).571C22, (C2×M4(2)).64C22, C42⋊C2.116C22, (C2×C4×D4).83C2, C4.85(C2×C4○D4), (C2×C4).99(C2×Q8), C2.23(C2×C8⋊C22), C2.56(C2×C22⋊Q8), (C2×C4).1436(C2×D4), (C2×C4).292(C4○D4), (C2×C4⋊C4).922C22, SmallGroup(128,1809)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.219D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=a2b2c3 >
Subgroups: 428 in 220 conjugacy classes, 102 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C23×C4, C22×D4, C23.37D4, C4⋊M4(2), M4(2)⋊C4, D4⋊Q8, D4⋊2Q8, D4.Q8, C2×C4×D4, C23.37C23, C42.219D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C8⋊C22, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C8⋊C22, D8⋊C22, C42.219D4
►On 32 points
(1 19 31 13)(2 14 32 20)(3 21 25 15)(4 16 26 22)(5 23 27 9)(6 10 28 24)(7 17 29 11)(8 12 30 18)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 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 8 31 30)(2 29 32 7)(3 6 25 28)(4 27 26 5)(9 16 23 22)(10 21 24 15)(11 14 17 20)(12 19 18 13)
G:=sub<Sym(32)| (1,19,31,13)(2,14,32,20)(3,21,25,15)(4,16,26,22)(5,23,27,9)(6,10,28,24)(7,17,29,11)(8,12,30,18), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32), (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), (1,8,31,30)(2,29,32,7)(3,6,25,28)(4,27,26,5)(9,16,23,22)(10,21,24,15)(11,14,17,20)(12,19,18,13)>;
G:=Group( (1,19,31,13)(2,14,32,20)(3,21,25,15)(4,16,26,22)(5,23,27,9)(6,10,28,24)(7,17,29,11)(8,12,30,18), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32), (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), (1,8,31,30)(2,29,32,7)(3,6,25,28)(4,27,26,5)(9,16,23,22)(10,21,24,15)(11,14,17,20)(12,19,18,13) );
G=PermutationGroup([[(1,19,31,13),(2,14,32,20),(3,21,25,15),(4,16,26,22),(5,23,27,9),(6,10,28,24),(7,17,29,11),(8,12,30,18)], [(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32)], [(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)], [(1,8,31,30),(2,29,32,7),(3,6,25,28),(4,27,26,5),(9,16,23,22),(10,21,24,15),(11,14,17,20),(12,19,18,13)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C8⋊C22 | D8⋊C22 |
| kernel | C42.219D4 | C23.37D4 | C4⋊M4(2) | M4(2)⋊C4 | D4⋊Q8 | D4⋊2Q8 | D4.Q8 | C2×C4×D4 | C23.37C23 | C42 | C22×C4 | C2×D4 | C2×C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.219D4 ►in GL6(𝔽17)
| 16 | 13 | 0 | 0 | 0 | 0 |
| 9 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 1 | 16 | 0 |
| 8 | 4 | 0 | 0 | 0 | 0 |
| 5 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 16 | 0 | 16 | 16 |
| 0 | 0 | 1 | 16 | 1 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 9 | 13 | 0 | 0 | 0 | 0 |
| 12 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 |
G:=sub<GL(6,GF(17))| [16,9,0,0,0,0,13,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0],[8,5,0,0,0,0,4,9,0,0,0,0,0,0,16,16,1,1,0,0,0,0,16,0,0,0,15,16,1,1,0,0,0,16,0,0],[9,12,0,0,0,0,13,8,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,16,1,1,0,0,0,1,0,0] >;
C42.219D4 in GAP, Magma, Sage, TeX
C_4^2._{219}D_4 % in TeX
G:=Group("C4^2.219D4"); // GroupNames label
G:=SmallGroup(128,1809);
// by ID
G=gap.SmallGroup(128,1809);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,2019,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations