p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.227D4, C42.343C23, D4⋊3(C4○D4), D8⋊C4⋊5C2, C4⋊D8⋊21C2, C4⋊C8⋊11C22, D4⋊2Q8⋊5C2, (C4×D4)⋊7C22, C4⋊Q8⋊59C22, C22⋊D8⋊15C2, C8⋊C4⋊2C22, C4⋊C4.62C23, (C2×C8).36C23, C4.Q8⋊12C22, C4⋊1D4⋊35C22, C42.6C4⋊1C2, (C2×C4).307C24, (C2×D8).57C22, (C2×D4).90C23, C23.672(C2×D4), (C22×C4).447D4, D4⋊C4⋊21C22, C4.102(C8⋊C22), C23.46D4⋊5C2, C22⋊C8.20C22, C4⋊D4.164C22, C22.47(C8⋊C22), C22.26C24⋊6C2, (C2×C42).834C22, C22.567(C22×D4), (C22×C4).1023C23, (C22×D4).575C22, C2.108(C22.19C24), (C2×C4×D4)⋊64C2, C4.192(C2×C4○D4), (C2×C4).496(C2×D4), C2.32(C2×C8⋊C22), (C2×C4⋊C4).936C22, SmallGroup(128,1841)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.227D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=dbd=a2b, dcd=a2c3 >
Subgroups: 532 in 244 conjugacy classes, 92 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, 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, C8⋊C4, C22⋊C8, D4⋊C4, C4⋊C8, C4.Q8, 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, C42.6C4, D8⋊C4, C22⋊D8, C4⋊D8, D4⋊2Q8, C23.46D4, C2×C4×D4, C22.26C24, C42.227D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, C22.19C24, C2×C8⋊C22, C42.227D4
►On 32 points
(1 9 5 13)(2 8 6 4)(3 11 7 15)(10 16 14 12)(17 27 21 31)(18 24 22 20)(19 29 23 25)(26 32 30 28)
(1 29 15 21)(2 26 16 18)(3 31 9 23)(4 28 10 20)(5 25 11 17)(6 30 12 22)(7 27 13 19)(8 32 14 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 10)(2 9)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)
G:=sub<Sym(32)| (1,9,5,13)(2,8,6,4)(3,11,7,15)(10,16,14,12)(17,27,21,31)(18,24,22,20)(19,29,23,25)(26,32,30,28), (1,29,15,21)(2,26,16,18)(3,31,9,23)(4,28,10,20)(5,25,11,17)(6,30,12,22)(7,27,13,19)(8,32,14,24), (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), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)>;
G:=Group( (1,9,5,13)(2,8,6,4)(3,11,7,15)(10,16,14,12)(17,27,21,31)(18,24,22,20)(19,29,23,25)(26,32,30,28), (1,29,15,21)(2,26,16,18)(3,31,9,23)(4,28,10,20)(5,25,11,17)(6,30,12,22)(7,27,13,19)(8,32,14,24), (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), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29) );
G=PermutationGroup([[(1,9,5,13),(2,8,6,4),(3,11,7,15),(10,16,14,12),(17,27,21,31),(18,24,22,20),(19,29,23,25),(26,32,30,28)], [(1,29,15,21),(2,26,16,18),(3,31,9,23),(4,28,10,20),(5,25,11,17),(6,30,12,22),(7,27,13,19),(8,32,14,24)], [(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)], [(1,10),(2,9),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(17,28),(18,27),(19,26),(20,25),(21,32),(22,31),(23,30),(24,29)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4N | 4O | 4P | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 | C8⋊C22 |
| kernel | C42.227D4 | C42.6C4 | D8⋊C4 | C22⋊D8 | C4⋊D8 | D4⋊2Q8 | C23.46D4 | C2×C4×D4 | C22.26C24 | C42 | C22×C4 | D4 | C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.227D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 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 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.227D4 in GAP, Magma, Sage, TeX
C_4^2._{227}D_4 % in TeX
G:=Group("C4^2.227D4"); // GroupNames label
G:=SmallGroup(128,1841);
// by ID
G=gap.SmallGroup(128,1841);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,521,80,4037,1027,124]);
// 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,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations