p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.293D4, C4.12- 1+4, C42.427C23, C4.202+ 1+4, (C2×C4)⋊5D8, C4⋊D8⋊9C2, C8⋊7D4⋊7C2, C4.74(C2×D8), C22.2(C2×D8), D4⋊Q8⋊10C2, C2.17(C22×D8), C4⋊C4.184C23, C4⋊C8.291C22, (C2×C8).170C23, (C2×C4).443C24, (C2×D8).27C22, (C22×C4).521D4, C23.402(C2×D4), C4⋊Q8.322C22, C2.D8.43C22, D4⋊C4.5C22, (C2×D4).186C23, (C4×D4).124C22, C4⋊D4.206C22, C4⋊1D4.174C22, (C22×C8).153C22, (C2×C42).900C22, C22.703(C22×D4), C2.68(D8⋊C22), (C22×C4).1576C23, C22.26C24⋊22C2, C2.62(C22.31C24), (C2×C4⋊C8)⋊24C2, (C2×C4).567(C2×D4), SmallGroup(128,1977)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.293D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, ac=ca, dad=ab2, cbc-1=a2b-1, dbd=b-1, dcd=a2c3 >
Subgroups: 484 in 218 conjugacy classes, 94 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, D4⋊C4, C4⋊C8, C2.D8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×D8, C2×C4○D4, C2×C4⋊C8, C4⋊D8, C8⋊7D4, D4⋊Q8, C22.26C24, C42.293D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C22×D8, D8⋊C22, C42.293D4
►On 64 points
(1 56 5 52)(2 49 6 53)(3 50 7 54)(4 51 8 55)(9 39 13 35)(10 40 14 36)(11 33 15 37)(12 34 16 38)(17 63 21 59)(18 64 22 60)(19 57 23 61)(20 58 24 62)(25 44 29 48)(26 45 30 41)(27 46 31 42)(28 47 32 43)
(1 19 47 14)(2 11 48 24)(3 21 41 16)(4 13 42 18)(5 23 43 10)(6 15 44 20)(7 17 45 12)(8 9 46 22)(25 62 49 33)(26 38 50 59)(27 64 51 35)(28 40 52 61)(29 58 53 37)(30 34 54 63)(31 60 55 39)(32 36 56 57)
(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 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 48)(25 33)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)
G:=sub<Sym(64)| (1,56,5,52)(2,49,6,53)(3,50,7,54)(4,51,8,55)(9,39,13,35)(10,40,14,36)(11,33,15,37)(12,34,16,38)(17,63,21,59)(18,64,22,60)(19,57,23,61)(20,58,24,62)(25,44,29,48)(26,45,30,41)(27,46,31,42)(28,47,32,43), (1,19,47,14)(2,11,48,24)(3,21,41,16)(4,13,42,18)(5,23,43,10)(6,15,44,20)(7,17,45,12)(8,9,46,22)(25,62,49,33)(26,38,50,59)(27,64,51,35)(28,40,52,61)(29,58,53,37)(30,34,54,63)(31,60,55,39)(32,36,56,57), (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), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,48)(25,33)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63)>;
G:=Group( (1,56,5,52)(2,49,6,53)(3,50,7,54)(4,51,8,55)(9,39,13,35)(10,40,14,36)(11,33,15,37)(12,34,16,38)(17,63,21,59)(18,64,22,60)(19,57,23,61)(20,58,24,62)(25,44,29,48)(26,45,30,41)(27,46,31,42)(28,47,32,43), (1,19,47,14)(2,11,48,24)(3,21,41,16)(4,13,42,18)(5,23,43,10)(6,15,44,20)(7,17,45,12)(8,9,46,22)(25,62,49,33)(26,38,50,59)(27,64,51,35)(28,40,52,61)(29,58,53,37)(30,34,54,63)(31,60,55,39)(32,36,56,57), (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), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,47)(18,46)(19,45)(20,44)(21,43)(22,42)(23,41)(24,48)(25,33)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63) );
G=PermutationGroup([[(1,56,5,52),(2,49,6,53),(3,50,7,54),(4,51,8,55),(9,39,13,35),(10,40,14,36),(11,33,15,37),(12,34,16,38),(17,63,21,59),(18,64,22,60),(19,57,23,61),(20,58,24,62),(25,44,29,48),(26,45,30,41),(27,46,31,42),(28,47,32,43)], [(1,19,47,14),(2,11,48,24),(3,21,41,16),(4,13,42,18),(5,23,43,10),(6,15,44,20),(7,17,45,12),(8,9,46,22),(25,62,49,33),(26,38,50,59),(27,64,51,35),(28,40,52,61),(29,58,53,37),(30,34,54,63),(31,60,55,39),(32,36,56,57)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13),(17,47),(18,46),(19,45),(20,44),(21,43),(22,42),(23,41),(24,48),(25,33),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,64),(56,63)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | 2+ 1+4 | 2- 1+4 | D8⋊C22 |
| kernel | C42.293D4 | C2×C4⋊C8 | C4⋊D8 | C8⋊7D4 | D4⋊Q8 | C22.26C24 | C42 | C22×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 |
Matrix representation of C42.293D4 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 11 | 0 |
| 0 | 0 | 16 | 16 | 3 | 3 |
| 0 | 0 | 8 | 11 | 1 | 1 |
| 0 | 0 | 3 | 1 | 15 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 16 |
| 0 | 0 | 6 | 6 | 1 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 15 | 15 |
| 0 | 0 | 15 | 9 | 0 | 5 |
| 0 | 0 | 2 | 0 | 0 | 12 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 16 |
| 0 | 0 | 11 | 11 | 16 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,8,3,0,0,2,16,11,1,0,0,11,3,1,15,0,0,0,3,1,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,16,0,6,0,0,2,16,11,6,0,0,0,0,0,1,0,0,0,0,16,0],[14,3,0,0,0,0,14,14,0,0,0,0,0,0,5,0,15,2,0,0,0,0,9,0,0,0,0,15,0,0,0,0,4,15,5,12],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,6,11,0,0,15,1,6,11,0,0,0,0,0,16,0,0,0,0,16,0] >;
C42.293D4 in GAP, Magma, Sage, TeX
C_4^2._{293}D_4 % in TeX
G:=Group("C4^2.293D4"); // GroupNames label
G:=SmallGroup(128,1977);
// by ID
G=gap.SmallGroup(128,1977);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d=b^-1,d*c*d=a^2*c^3>;
// generators/relations