p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.302D4, C42.436C23, C4.102- 1+4, C4.292+ 1+4, C8⋊D4⋊21C2, C4⋊SD16⋊14C2, D4⋊Q8⋊31C2, C4.Q16⋊31C2, C4⋊C8.82C22, (C2×C8).78C23, D4.D4⋊14C2, C4⋊C4.193C23, (C2×C4).452C24, C23.309(C2×D4), (C22×C4).529D4, C4⋊Q8.330C22, C4.107(C8⋊C22), C4⋊M4(2)⋊13C2, (C4×D4).132C22, (C2×D4).194C23, (C4×Q8).128C22, (C2×Q8).181C23, C2.D8.113C22, D4⋊C4.59C22, C4⋊1D4.179C22, C4⋊D4.214C22, C4.102(C8.C22), (C2×C42).909C22, Q8⋊C4.56C22, (C2×SD16).42C22, C22.712(C22×D4), C22⋊Q8.218C22, (C22×C4).1107C23, (C2×M4(2)).90C22, C23.37C23⋊27C2, C22.26C24.49C2, C2.71(C22.31C24), (C2×C4).576(C2×D4), C2.69(C2×C8⋊C22), C2.68(C2×C8.C22), SmallGroup(128,1986)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.302D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=c3 >
Subgroups: 388 in 193 conjugacy classes, 88 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×SD16, C2×C4○D4, C4⋊M4(2), C4⋊SD16, D4.D4, C8⋊D4, D4⋊Q8, C4.Q16, C22.26C24, C23.37C23, C42.302D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C8.C22, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C8⋊C22, C2×C8.C22, C42.302D4
Character table of C42.302D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 64 points
(1 59 5 63)(2 64 6 60)(3 61 7 57)(4 58 8 62)(9 36 13 40)(10 33 14 37)(11 38 15 34)(12 35 16 39)(17 26 21 30)(18 31 22 27)(19 28 23 32)(20 25 24 29)(41 50 45 54)(42 55 46 51)(43 52 47 56)(44 49 48 53)
(1 31 51 11)(2 12 52 32)(3 25 53 13)(4 14 54 26)(5 27 55 15)(6 16 56 28)(7 29 49 9)(8 10 50 30)(17 62 33 45)(18 46 34 63)(19 64 35 47)(20 48 36 57)(21 58 37 41)(22 42 38 59)(23 60 39 43)(24 44 40 61)
(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 4 5 8)(2 7 6 3)(9 16 13 12)(10 11 14 15)(17 38 21 34)(18 33 22 37)(19 36 23 40)(20 39 24 35)(25 32 29 28)(26 27 30 31)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)
G:=sub<Sym(64)| (1,59,5,63)(2,64,6,60)(3,61,7,57)(4,58,8,62)(9,36,13,40)(10,33,14,37)(11,38,15,34)(12,35,16,39)(17,26,21,30)(18,31,22,27)(19,28,23,32)(20,25,24,29)(41,50,45,54)(42,55,46,51)(43,52,47,56)(44,49,48,53), (1,31,51,11)(2,12,52,32)(3,25,53,13)(4,14,54,26)(5,27,55,15)(6,16,56,28)(7,29,49,9)(8,10,50,30)(17,62,33,45)(18,46,34,63)(19,64,35,47)(20,48,36,57)(21,58,37,41)(22,42,38,59)(23,60,39,43)(24,44,40,61), (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), (1,4,5,8)(2,7,6,3)(9,16,13,12)(10,11,14,15)(17,38,21,34)(18,33,22,37)(19,36,23,40)(20,39,24,35)(25,32,29,28)(26,27,30,31)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55)>;
G:=Group( (1,59,5,63)(2,64,6,60)(3,61,7,57)(4,58,8,62)(9,36,13,40)(10,33,14,37)(11,38,15,34)(12,35,16,39)(17,26,21,30)(18,31,22,27)(19,28,23,32)(20,25,24,29)(41,50,45,54)(42,55,46,51)(43,52,47,56)(44,49,48,53), (1,31,51,11)(2,12,52,32)(3,25,53,13)(4,14,54,26)(5,27,55,15)(6,16,56,28)(7,29,49,9)(8,10,50,30)(17,62,33,45)(18,46,34,63)(19,64,35,47)(20,48,36,57)(21,58,37,41)(22,42,38,59)(23,60,39,43)(24,44,40,61), (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), (1,4,5,8)(2,7,6,3)(9,16,13,12)(10,11,14,15)(17,38,21,34)(18,33,22,37)(19,36,23,40)(20,39,24,35)(25,32,29,28)(26,27,30,31)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55) );
G=PermutationGroup([[(1,59,5,63),(2,64,6,60),(3,61,7,57),(4,58,8,62),(9,36,13,40),(10,33,14,37),(11,38,15,34),(12,35,16,39),(17,26,21,30),(18,31,22,27),(19,28,23,32),(20,25,24,29),(41,50,45,54),(42,55,46,51),(43,52,47,56),(44,49,48,53)], [(1,31,51,11),(2,12,52,32),(3,25,53,13),(4,14,54,26),(5,27,55,15),(6,16,56,28),(7,29,49,9),(8,10,50,30),(17,62,33,45),(18,46,34,63),(19,64,35,47),(20,48,36,57),(21,58,37,41),(22,42,38,59),(23,60,39,43),(24,44,40,61)], [(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)], [(1,4,5,8),(2,7,6,3),(9,16,13,12),(10,11,14,15),(17,38,21,34),(18,33,22,37),(19,36,23,40),(20,39,24,35),(25,32,29,28),(26,27,30,31),(41,63,45,59),(42,58,46,62),(43,61,47,57),(44,64,48,60),(49,56,53,52),(50,51,54,55)]])
Matrix representation of C42.302D4 ►in GL8(𝔽17)
| 1 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 | 8 | 8 |
| 0 | 0 | 0 | 0 | 15 | 15 | 9 | 8 |
| 0 | 0 | 0 | 0 | 8 | 8 | 2 | 15 |
| 0 | 0 | 0 | 0 | 9 | 8 | 2 | 2 |
| 4 | 0 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 | 8 | 8 |
| 0 | 0 | 0 | 0 | 2 | 2 | 8 | 9 |
| 0 | 0 | 0 | 0 | 9 | 9 | 15 | 2 |
| 0 | 0 | 0 | 0 | 9 | 8 | 2 | 2 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,9,13,13,13,0,0,0,0,0,0,0,0,15,15,8,9,0,0,0,0,2,15,8,8,0,0,0,0,8,9,2,2,0,0,0,0,8,8,15,2],[4,0,4,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,9,13,13,13,0,0,0,0,0,0,0,0,15,2,9,9,0,0,0,0,2,2,9,8,0,0,0,0,8,8,15,2,0,0,0,0,8,9,2,2] >;
C42.302D4 in GAP, Magma, Sage, TeX
C_4^2._{302}D_4 % in TeX
G:=Group("C4^2.302D4"); // GroupNames label
G:=SmallGroup(128,1986);
// by ID
G=gap.SmallGroup(128,1986);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,891,675,80,4037,1027,124]);
// 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,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
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