p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.303D4, C42.437C23, C4.112- 1+4, C4.302+ 1+4, Q8⋊Q8⋊14C2, C4⋊2Q16⋊31C2, C4⋊C8.83C22, (C2×C8).79C23, C8.D4.6C2, C4⋊C4.194C23, (C2×C4).453C24, (C22×C4).530D4, C23.310(C2×D4), C4⋊Q8.331C22, C4.Q8.48C22, C4⋊M4(2).3C2, (C2×Q16).75C22, (C4×Q8).129C22, (C2×Q8).182C23, C4.103(C8.C22), (C2×C42).910C22, Q8⋊C4.57C22, C22.713(C22×D4), C22⋊Q8.219C22, (C22×C4).1108C23, (C2×M4(2)).91C22, C23.37C23.44C2, C2.72(C22.31C24), (C2×C4).577(C2×D4), C2.69(C2×C8.C22), SmallGroup(128,1987)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.303D4
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=a2c3 >
Subgroups: 292 in 171 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, Q8⋊C4, C4⋊C8, C4.Q8, C2×C42, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×M4(2), C2×Q16, C4⋊M4(2), C4⋊2Q16, C8.D4, Q8⋊Q8, C23.37C23, C42.303D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C8.C22, C42.303D4
Character table of C42.303D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 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 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 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 |
| ρ23 | 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 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 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
Generators in S64
(1 3 5 7)(2 8 6 4)(9 38 13 34)(10 35 14 39)(11 40 15 36)(12 37 16 33)(17 19 21 23)(18 24 22 20)(25 48 29 44)(26 45 30 41)(27 42 31 46)(28 47 32 43)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)
(1 54 59 17)(2 18 60 55)(3 56 61 19)(4 20 62 49)(5 50 63 21)(6 22 64 51)(7 52 57 23)(8 24 58 53)(9 41 36 32)(10 25 37 42)(11 43 38 26)(12 27 39 44)(13 45 40 28)(14 29 33 46)(15 47 34 30)(16 31 35 48)
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 51 13 55)(10 50 14 54)(11 49 15 53)(12 56 16 52)(17 37 21 33)(18 36 22 40)(19 35 23 39)(20 34 24 38)(41 64 45 60)(42 63 46 59)(43 62 47 58)(44 61 48 57)
G:=sub<Sym(64)| (1,3,5,7)(2,8,6,4)(9,38,13,34)(10,35,14,39)(11,40,15,36)(12,37,16,33)(17,19,21,23)(18,24,22,20)(25,48,29,44)(26,45,30,41)(27,42,31,46)(28,47,32,43)(49,55,53,51)(50,52,54,56)(57,59,61,63)(58,64,62,60), (1,54,59,17)(2,18,60,55)(3,56,61,19)(4,20,62,49)(5,50,63,21)(6,22,64,51)(7,52,57,23)(8,24,58,53)(9,41,36,32)(10,25,37,42)(11,43,38,26)(12,27,39,44)(13,45,40,28)(14,29,33,46)(15,47,34,30)(16,31,35,48), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57)>;
G:=Group( (1,3,5,7)(2,8,6,4)(9,38,13,34)(10,35,14,39)(11,40,15,36)(12,37,16,33)(17,19,21,23)(18,24,22,20)(25,48,29,44)(26,45,30,41)(27,42,31,46)(28,47,32,43)(49,55,53,51)(50,52,54,56)(57,59,61,63)(58,64,62,60), (1,54,59,17)(2,18,60,55)(3,56,61,19)(4,20,62,49)(5,50,63,21)(6,22,64,51)(7,52,57,23)(8,24,58,53)(9,41,36,32)(10,25,37,42)(11,43,38,26)(12,27,39,44)(13,45,40,28)(14,29,33,46)(15,47,34,30)(16,31,35,48), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,51,13,55)(10,50,14,54)(11,49,15,53)(12,56,16,52)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(41,64,45,60)(42,63,46,59)(43,62,47,58)(44,61,48,57) );
G=PermutationGroup([[(1,3,5,7),(2,8,6,4),(9,38,13,34),(10,35,14,39),(11,40,15,36),(12,37,16,33),(17,19,21,23),(18,24,22,20),(25,48,29,44),(26,45,30,41),(27,42,31,46),(28,47,32,43),(49,55,53,51),(50,52,54,56),(57,59,61,63),(58,64,62,60)], [(1,54,59,17),(2,18,60,55),(3,56,61,19),(4,20,62,49),(5,50,63,21),(6,22,64,51),(7,52,57,23),(8,24,58,53),(9,41,36,32),(10,25,37,42),(11,43,38,26),(12,27,39,44),(13,45,40,28),(14,29,33,46),(15,47,34,30),(16,31,35,48)], [(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)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,51,13,55),(10,50,14,54),(11,49,15,53),(12,56,16,52),(17,37,21,33),(18,36,22,40),(19,35,23,39),(20,34,24,38),(41,64,45,60),(42,63,46,59),(43,62,47,58),(44,61,48,57)]])
Matrix representation of C42.303D4 ►in GL8(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 8 | 0 | 0 | 0 | 0 |
| 11 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
G:=sub<GL(8,GF(17))| [16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,9,0,0,0,0,0,0,13,1,0,0,0,0,0,0,0,0,16,9,0,0,0,0,0,0,13,1],[0,0,11,1,0,0,0,0,0,0,14,6,0,0,0,0,9,10,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,4],[6,16,0,0,0,0,0,0,3,11,0,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,13,0,0,0,0,13,0,0,0,0,0,0,0,1,4,0,0] >;
C42.303D4 in GAP, Magma, Sage, TeX
C_4^2._{303}D_4 % in TeX
G:=Group("C4^2.303D4"); // GroupNames label
G:=SmallGroup(128,1987);
// by ID
G=gap.SmallGroup(128,1987);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,891,352,675,80,4037,1027,124]);
// 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=a^2*c^3>;
// generators/relations
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