p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.449D4, C42.336C23, C4○D4.9Q8, Q8.4(C2×Q8), D4.4(C2×Q8), D4.Q8⋊13C2, Q8.Q8⋊13C2, C4⋊C4.43C23, C4⋊C8.44C22, (C2×C8).27C23, C4.31(C22×Q8), C4⋊M4(2)⋊9C2, (C2×C4).278C24, (C22×C4).433D4, C23.660(C2×D4), C4.90(C22⋊Q8), C4.Q8.10C22, C2.D8.81C22, (C2×D4).396C23, (C4×D4).318C22, (C2×Q8).367C23, (C4×Q8).299C22, M4(2)⋊C4⋊18C2, D4⋊C4.25C22, (C2×C42).824C22, (C22×C4).997C23, Q8⋊C4.26C22, C23.36D4.2C2, C22.538(C22×D4), C22.55(C22⋊Q8), C2.20(D8⋊C22), (C2×M4(2)).67C22, C42.C2.104C22, C42⋊C2.315C22, C4.88(C2×C4○D4), (C4×C4○D4).25C2, (C2×C4).102(C2×Q8), C2.59(C2×C22⋊Q8), (C2×C4).1215(C2×D4), (C2×C42.C2)⋊32C2, (C2×C4).295(C4○D4), (C2×C4⋊C4).604C22, (C2×C4○D4).309C22, SmallGroup(128,1812)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.449D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2c3 >
Subgroups: 324 in 192 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, 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, C4○D4, C4○D4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C42.C2, C42.C2, C2×M4(2), C2×C4○D4, C23.36D4, C4⋊M4(2), M4(2)⋊C4, D4.Q8, Q8.Q8, C4×C4○D4, C2×C42.C2, C42.449D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D8⋊C22, C42.449D4
►On 64 points
(1 49 10 63)(2 60 11 54)(3 51 12 57)(4 62 13 56)(5 53 14 59)(6 64 15 50)(7 55 16 61)(8 58 9 52)(17 41 37 28)(18 25 38 46)(19 43 39 30)(20 27 40 48)(21 45 33 32)(22 29 34 42)(23 47 35 26)(24 31 36 44)
(1 26 5 30)(2 31 6 27)(3 28 7 32)(4 25 8 29)(9 42 13 46)(10 47 14 43)(11 44 15 48)(12 41 16 45)(17 55 21 51)(18 52 22 56)(19 49 23 53)(20 54 24 50)(33 57 37 61)(34 62 38 58)(35 59 39 63)(36 64 40 60)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 13 14 8)(2 7 15 12)(3 11 16 6)(4 5 9 10)(17 36 33 20)(18 19 34 35)(21 40 37 24)(22 23 38 39)(25 26 42 43)(27 32 44 41)(28 48 45 31)(29 30 46 47)(49 58 59 56)(50 55 60 57)(51 64 61 54)(52 53 62 63)
G:=sub<Sym(64)| (1,49,10,63)(2,60,11,54)(3,51,12,57)(4,62,13,56)(5,53,14,59)(6,64,15,50)(7,55,16,61)(8,58,9,52)(17,41,37,28)(18,25,38,46)(19,43,39,30)(20,27,40,48)(21,45,33,32)(22,29,34,42)(23,47,35,26)(24,31,36,44), (1,26,5,30)(2,31,6,27)(3,28,7,32)(4,25,8,29)(9,42,13,46)(10,47,14,43)(11,44,15,48)(12,41,16,45)(17,55,21,51)(18,52,22,56)(19,49,23,53)(20,54,24,50)(33,57,37,61)(34,62,38,58)(35,59,39,63)(36,64,40,60), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,13,14,8)(2,7,15,12)(3,11,16,6)(4,5,9,10)(17,36,33,20)(18,19,34,35)(21,40,37,24)(22,23,38,39)(25,26,42,43)(27,32,44,41)(28,48,45,31)(29,30,46,47)(49,58,59,56)(50,55,60,57)(51,64,61,54)(52,53,62,63)>;
G:=Group( (1,49,10,63)(2,60,11,54)(3,51,12,57)(4,62,13,56)(5,53,14,59)(6,64,15,50)(7,55,16,61)(8,58,9,52)(17,41,37,28)(18,25,38,46)(19,43,39,30)(20,27,40,48)(21,45,33,32)(22,29,34,42)(23,47,35,26)(24,31,36,44), (1,26,5,30)(2,31,6,27)(3,28,7,32)(4,25,8,29)(9,42,13,46)(10,47,14,43)(11,44,15,48)(12,41,16,45)(17,55,21,51)(18,52,22,56)(19,49,23,53)(20,54,24,50)(33,57,37,61)(34,62,38,58)(35,59,39,63)(36,64,40,60), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,13,14,8)(2,7,15,12)(3,11,16,6)(4,5,9,10)(17,36,33,20)(18,19,34,35)(21,40,37,24)(22,23,38,39)(25,26,42,43)(27,32,44,41)(28,48,45,31)(29,30,46,47)(49,58,59,56)(50,55,60,57)(51,64,61,54)(52,53,62,63) );
G=PermutationGroup([[(1,49,10,63),(2,60,11,54),(3,51,12,57),(4,62,13,56),(5,53,14,59),(6,64,15,50),(7,55,16,61),(8,58,9,52),(17,41,37,28),(18,25,38,46),(19,43,39,30),(20,27,40,48),(21,45,33,32),(22,29,34,42),(23,47,35,26),(24,31,36,44)], [(1,26,5,30),(2,31,6,27),(3,28,7,32),(4,25,8,29),(9,42,13,46),(10,47,14,43),(11,44,15,48),(12,41,16,45),(17,55,21,51),(18,52,22,56),(19,49,23,53),(20,54,24,50),(33,57,37,61),(34,62,38,58),(35,59,39,63),(36,64,40,60)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,13,14,8),(2,7,15,12),(3,11,16,6),(4,5,9,10),(17,36,33,20),(18,19,34,35),(21,40,37,24),(22,23,38,39),(25,26,42,43),(27,32,44,41),(28,48,45,31),(29,30,46,47),(49,58,59,56),(50,55,60,57),(51,64,61,54),(52,53,62,63)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 8A | 8B | 8C | 8D |
| order | 1 | 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 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | D8⋊C22 |
| kernel | C42.449D4 | C23.36D4 | C4⋊M4(2) | M4(2)⋊C4 | D4.Q8 | Q8.Q8 | C4×C4○D4 | C2×C42.C2 | C42 | C22×C4 | C4○D4 | C2×C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C42.449D4 ►in GL6(𝔽17)
| 4 | 15 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 16 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 5 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 12 | 7 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,15,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,13,0,0,0,0,0,0,13,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[4,16,0,0,0,0,0,13,0,0,0,0,0,0,7,5,0,0,0,0,7,0,0,0,0,0,0,0,10,12,0,0,0,0,10,0],[13,1,0,0,0,0,0,4,0,0,0,0,0,0,7,5,0,0,0,0,7,10,0,0,0,0,0,0,10,12,0,0,0,0,10,7] >;
C42.449D4 in GAP, Magma, Sage, TeX
C_4^2._{449}D_4 % in TeX
G:=Group("C4^2.449D4"); // GroupNames label
G:=SmallGroup(128,1812);
// by ID
G=gap.SmallGroup(128,1812);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,2019,248,4037,1027,124]);
// 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,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^3>;
// generators/relations