p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.448D4, C42.335C23, C4○D4⋊4Q8, D4⋊3(C2×Q8), Q8⋊3(C2×Q8), Q8⋊Q8⋊2C2, D4⋊2Q8⋊2C2, C4.Q16⋊19C2, D4⋊Q8⋊19C2, C4⋊C8.43C22, C4⋊C4.42C23, (C2×C8).26C23, C4.30(C22×Q8), C4⋊M4(2)⋊8C2, (C2×C4).277C24, C4.Q8.9C22, (C22×C4).432D4, C23.659(C2×D4), C4⋊Q8.262C22, C4.89(C22⋊Q8), C4.124(C8⋊C22), C2.D8.80C22, (C4×D4).317C22, (C2×D4).395C23, (C2×Q8).366C23, (C4×Q8).298C22, M4(2)⋊C4⋊17C2, D4⋊C4.24C22, C4.118(C8.C22), (C2×C42).823C22, (C22×C4).996C23, Q8⋊C4.25C22, C23.36D4.1C2, C22.537(C22×D4), C22.54(C22⋊Q8), (C2×M4(2)).66C22, C42⋊C2.314C22, (C2×C4⋊Q8)⋊33C2, C4.87(C2×C4○D4), (C4×C4○D4).24C2, C2.24(C2×C8⋊C22), (C2×C4).101(C2×Q8), C2.58(C2×C22⋊Q8), (C2×C4).1438(C2×D4), C2.24(C2×C8.C22), (C2×C4).294(C4○D4), (C2×C4⋊C4).603C22, (C2×C4○D4).308C22, SmallGroup(128,1811)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.448D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2b2c3 >
Subgroups: 372 in 208 conjugacy classes, 104 normal (36 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, 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, C4⋊Q8, C4⋊Q8, C2×M4(2), C22×Q8, C2×C4○D4, C23.36D4, C4⋊M4(2), M4(2)⋊C4, D4⋊Q8, Q8⋊Q8, D4⋊2Q8, C4.Q16, C4×C4○D4, C2×C4⋊Q8, C42.448D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C8⋊C22, C8.C22, C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C2×C8⋊C22, C2×C8.C22, C42.448D4
►On 64 points
(1 55 10 63)(2 64 11 56)(3 49 12 57)(4 58 13 50)(5 51 14 59)(6 60 15 52)(7 53 16 61)(8 62 9 54)(17 43 39 30)(18 31 40 44)(19 45 33 32)(20 25 34 46)(21 47 35 26)(22 27 36 48)(23 41 37 28)(24 29 38 42)
(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 9 10 8)(2 7 11 16)(3 15 12 6)(4 5 13 14)(17 38 39 24)(18 23 40 37)(19 36 33 22)(20 21 34 35)(25 26 46 47)(27 32 48 45)(28 44 41 31)(29 30 42 43)(49 60 57 52)(50 51 58 59)(53 64 61 56)(54 55 62 63)
G:=sub<Sym(64)| (1,55,10,63)(2,64,11,56)(3,49,12,57)(4,58,13,50)(5,51,14,59)(6,60,15,52)(7,53,16,61)(8,62,9,54)(17,43,39,30)(18,31,40,44)(19,45,33,32)(20,25,34,46)(21,47,35,26)(22,27,36,48)(23,41,37,28)(24,29,38,42), (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), (1,9,10,8)(2,7,11,16)(3,15,12,6)(4,5,13,14)(17,38,39,24)(18,23,40,37)(19,36,33,22)(20,21,34,35)(25,26,46,47)(27,32,48,45)(28,44,41,31)(29,30,42,43)(49,60,57,52)(50,51,58,59)(53,64,61,56)(54,55,62,63)>;
G:=Group( (1,55,10,63)(2,64,11,56)(3,49,12,57)(4,58,13,50)(5,51,14,59)(6,60,15,52)(7,53,16,61)(8,62,9,54)(17,43,39,30)(18,31,40,44)(19,45,33,32)(20,25,34,46)(21,47,35,26)(22,27,36,48)(23,41,37,28)(24,29,38,42), (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), (1,9,10,8)(2,7,11,16)(3,15,12,6)(4,5,13,14)(17,38,39,24)(18,23,40,37)(19,36,33,22)(20,21,34,35)(25,26,46,47)(27,32,48,45)(28,44,41,31)(29,30,42,43)(49,60,57,52)(50,51,58,59)(53,64,61,56)(54,55,62,63) );
G=PermutationGroup([[(1,55,10,63),(2,64,11,56),(3,49,12,57),(4,58,13,50),(5,51,14,59),(6,60,15,52),(7,53,16,61),(8,62,9,54),(17,43,39,30),(18,31,40,44),(19,45,33,32),(20,25,34,46),(21,47,35,26),(22,27,36,48),(23,41,37,28),(24,29,38,42)], [(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)], [(1,9,10,8),(2,7,11,16),(3,15,12,6),(4,5,13,14),(17,38,39,24),(18,23,40,37),(19,36,33,22),(20,21,34,35),(25,26,46,47),(27,32,48,45),(28,44,41,31),(29,30,42,43),(49,60,57,52),(50,51,58,59),(53,64,61,56),(54,55,62,63)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | C4○D4 | C8⋊C22 | C8.C22 |
| kernel | C42.448D4 | C23.36D4 | C4⋊M4(2) | M4(2)⋊C4 | D4⋊Q8 | Q8⋊Q8 | D4⋊2Q8 | C4.Q16 | C4×C4○D4 | C2×C4⋊Q8 | C42 | C22×C4 | C4○D4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 |
Matrix representation of C42.448D4 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 16 | 1 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 16 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 10 | 10 |
| 0 | 0 | 5 | 5 | 0 | 7 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 5 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 10 | 10 |
| 0 | 0 | 12 | 12 | 0 | 10 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 5 | 10 |
G:=sub<GL(6,GF(17))| [13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,16,1,16,0,0,0,16,0,16,0,0,16,1,0,0,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16,1,0,0,0,1,0,1,0,0,1,16,0,0,0,0,0,15,0,16],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,12,5,5,12,0,0,12,5,12,0,0,0,10,0,0,5,0,0,10,7,0,0],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,12,12,5,12,0,0,12,12,12,0,0,0,10,0,0,5,0,0,10,10,0,10] >;
C42.448D4 in GAP, Magma, Sage, TeX
C_4^2._{448}D_4 % in TeX
G:=Group("C4^2.448D4"); // GroupNames label
G:=SmallGroup(128,1811);
// by ID
G=gap.SmallGroup(128,1811);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,2019,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations