p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.228D4, C42.344C23, C4⋊C8⋊12C22, C4⋊Q8⋊60C22, (C4×Q8)⋊7C22, C8⋊C4⋊3C22, D4⋊Q8⋊22C2, D4.5(C4○D4), D4.D4⋊5C2, C4⋊C4.63C23, (C2×C8).37C23, C2.D8⋊23C22, SD16⋊C4⋊5C2, C42.6C4⋊2C2, (C2×C4).308C24, C22⋊SD16.1C2, C23.673(C2×D4), (C22×C4).448D4, (C2×Q8).76C23, C4.103(C8⋊C22), Q8⋊C4⋊23C22, (C2×D4).403C23, (C4×D4).321C22, C22⋊C8.21C22, D4⋊C4.29C22, C23.48D4⋊15C2, (C2×C42).835C22, (C2×SD16).10C22, C22.568(C22×D4), C22⋊Q8.169C22, (C22×C4).1024C23, C23.37C23⋊6C2, (C22×D4).576C22, C22.36(C8.C22), C2.109(C22.19C24), (C2×C4×D4).85C2, C4.193(C2×C4○D4), (C2×C4).497(C2×D4), C2.33(C2×C8⋊C22), C2.32(C2×C8.C22), (C2×C4⋊C4).937C22, SmallGroup(128,1842)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.228D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=c3 >
Subgroups: 436 in 222 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, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16, C23×C4, C22×D4, C42.6C4, SD16⋊C4, C22⋊SD16, D4.D4, D4⋊Q8, C23.48D4, C2×C4×D4, C23.37C23, C42.228D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C8.C22, C22×D4, C2×C4○D4, C22.19C24, C2×C8⋊C22, C2×C8.C22, C42.228D4
►On 32 points
(1 13 5 9)(2 4 6 8)(3 15 7 11)(10 12 14 16)(17 19 21 23)(18 31 22 27)(20 25 24 29)(26 28 30 32)
(1 27 15 20)(2 32 16 17)(3 29 9 22)(4 26 10 19)(5 31 11 24)(6 28 12 21)(7 25 13 18)(8 30 14 23)
(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 14 5 10)(2 9 6 13)(3 12 7 16)(4 15 8 11)(17 25 21 29)(18 28 22 32)(19 31 23 27)(20 26 24 30)
G:=sub<Sym(32)| (1,13,5,9)(2,4,6,8)(3,15,7,11)(10,12,14,16)(17,19,21,23)(18,31,22,27)(20,25,24,29)(26,28,30,32), (1,27,15,20)(2,32,16,17)(3,29,9,22)(4,26,10,19)(5,31,11,24)(6,28,12,21)(7,25,13,18)(8,30,14,23), (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), (1,14,5,10)(2,9,6,13)(3,12,7,16)(4,15,8,11)(17,25,21,29)(18,28,22,32)(19,31,23,27)(20,26,24,30)>;
G:=Group( (1,13,5,9)(2,4,6,8)(3,15,7,11)(10,12,14,16)(17,19,21,23)(18,31,22,27)(20,25,24,29)(26,28,30,32), (1,27,15,20)(2,32,16,17)(3,29,9,22)(4,26,10,19)(5,31,11,24)(6,28,12,21)(7,25,13,18)(8,30,14,23), (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), (1,14,5,10)(2,9,6,13)(3,12,7,16)(4,15,8,11)(17,25,21,29)(18,28,22,32)(19,31,23,27)(20,26,24,30) );
G=PermutationGroup([[(1,13,5,9),(2,4,6,8),(3,15,7,11),(10,12,14,16),(17,19,21,23),(18,31,22,27),(20,25,24,29),(26,28,30,32)], [(1,27,15,20),(2,32,16,17),(3,29,9,22),(4,26,10,19),(5,31,11,24),(6,28,12,21),(7,25,13,18),(8,30,14,23)], [(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)], [(1,14,5,10),(2,9,6,13),(3,12,7,16),(4,15,8,11),(17,25,21,29),(18,28,22,32),(19,31,23,27),(20,26,24,30)]])
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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.228D4 | C42.6C4 | SD16⋊C4 | C22⋊SD16 | D4.D4 | D4⋊Q8 | C23.48D4 | C2×C4×D4 | C23.37C23 | C42 | C22×C4 | D4 | C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.228D4 ►in GL6(𝔽17)
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 14 | 8 | 1 | 2 |
| 0 | 0 | 7 | 3 | 16 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 15 | 16 | 0 |
| 0 | 0 | 4 | 2 | 0 | 16 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 2 | 0 |
| 0 | 0 | 12 | 0 | 15 | 15 |
| 0 | 0 | 15 | 7 | 10 | 2 |
| 0 | 0 | 13 | 13 | 2 | 15 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 15 | 15 | 0 |
| 0 | 0 | 4 | 2 | 0 | 15 |
| 0 | 0 | 3 | 7 | 9 | 2 |
| 0 | 0 | 3 | 7 | 13 | 15 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,16,1,0,0,0,0,0,0,1,16,14,7,0,0,2,16,8,3,0,0,0,0,1,16,0,0,0,0,2,16],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,8,4,0,0,0,1,15,2,0,0,0,0,16,0,0,0,0,0,0,16],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,9,12,15,13,0,0,2,0,7,13,0,0,2,15,10,2,0,0,0,15,2,15],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,8,4,3,3,0,0,15,2,7,7,0,0,15,0,9,13,0,0,0,15,2,15] >;
C42.228D4 in GAP, Magma, Sage, TeX
C_4^2._{228}D_4 % in TeX
G:=Group("C4^2.228D4"); // GroupNames label
G:=SmallGroup(128,1842);
// by ID
G=gap.SmallGroup(128,1842);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,521,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations