p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.221D4, C42.337C23, (C4×D8)⋊3C2, (C2×C4)⋊11D8, (C4×C8)⋊7C22, C4.71(C2×D8), D4⋊1(C4○D4), C4○2(C4⋊D8), C4⋊D8⋊46C2, C4⋊C8⋊59C22, (C4×D4)⋊5C22, C4⋊Q8⋊57C22, C4○2(C22⋊D8), C22⋊D8⋊36C2, C2.9(C22×D8), C4○2(D4⋊Q8), D4⋊Q8⋊49C2, C4⋊C4.53C23, C22.22(C2×D8), C2.D8⋊56C22, C4⋊1D4⋊34C22, (C2×C4).298C24, (C2×C8).146C23, C23.666(C2×D4), (C22×C4).801D4, D4⋊C4⋊67C22, (C2×D4).400C23, (C2×D8).121C22, C4○2(C22.D8), C22.D8⋊38C2, C42.12C4⋊25C2, C4⋊D4.158C22, C22⋊C8.173C22, (C2×C42).825C22, C22.26C24⋊4C2, C22.558(C22×D4), C2.25(D8⋊C22), (C22×C4).1014C23, (C22×D4).572C22, C2.99(C22.19C24), (C2×C4×D4)⋊62C2, (C2×C4)○(D4⋊Q8), C4.183(C2×C4○D4), (C2×C4).1579(C2×D4), (C2×C4⋊C4).930C22, SmallGroup(128,1832)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.221D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1b2, dad=ab2, bc=cb, dbd=a2b, dcd=a2c3 >
Subgroups: 532 in 246 conjugacy classes, 98 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, 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.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×D8, C23×C4, C22×D4, C2×C4○D4, C42.12C4, C4×D8, C22⋊D8, C4⋊D8, D4⋊Q8, C22.D8, C2×C4×D4, C22.26C24, C42.221D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C24, C2×D8, C22×D4, C2×C4○D4, C22.19C24, C22×D8, D8⋊C22, C42.221D4
►On 32 points
(1 29 5 25)(2 8 6 4)(3 31 7 27)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 24 22 20)(26 32 30 28)
(1 11 27 17)(2 12 28 18)(3 13 29 19)(4 14 30 20)(5 15 31 21)(6 16 32 22)(7 9 25 23)(8 10 26 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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 32)(8 31)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)
G:=sub<Sym(32)| (1,29,5,25)(2,8,6,4)(3,31,7,27)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,24,22,20)(26,32,30,28), (1,11,27,17)(2,12,28,18)(3,13,29,19)(4,14,30,20)(5,15,31,21)(6,16,32,22)(7,9,25,23)(8,10,26,24), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,32)(8,31)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)>;
G:=Group( (1,29,5,25)(2,8,6,4)(3,31,7,27)(9,17,13,21)(10,16,14,12)(11,19,15,23)(18,24,22,20)(26,32,30,28), (1,11,27,17)(2,12,28,18)(3,13,29,19)(4,14,30,20)(5,15,31,21)(6,16,32,22)(7,9,25,23)(8,10,26,24), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,32)(8,31)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19) );
G=PermutationGroup([[(1,29,5,25),(2,8,6,4),(3,31,7,27),(9,17,13,21),(10,16,14,12),(11,19,15,23),(18,24,22,20),(26,32,30,28)], [(1,11,27,17),(2,12,28,18),(3,13,29,19),(4,14,30,20),(5,15,31,21),(6,16,32,22),(7,9,25,23),(8,10,26,24)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,32),(8,31),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4P | 4Q | 4R | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 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 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | C4○D4 | D8⋊C22 |
| kernel | C42.221D4 | C42.12C4 | C4×D8 | C22⋊D8 | C4⋊D8 | D4⋊Q8 | C22.D8 | C2×C4×D4 | C22.26C24 | C42 | C22×C4 | C2×C4 | D4 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 8 | 2 |
Matrix representation of C42.221D4 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 3 | 3 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 |
| 0 | 0 | 14 | 14 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,0,16,0,0,1,0],[0,1,0,0,16,0,0,0,0,0,3,3,0,0,14,3],[0,1,0,0,1,0,0,0,0,0,3,14,0,0,14,14] >;
C42.221D4 in GAP, Magma, Sage, TeX
C_4^2._{221}D_4 % in TeX
G:=Group("C4^2.221D4"); // GroupNames label
G:=SmallGroup(128,1832);
// by ID
G=gap.SmallGroup(128,1832);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations