p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.31D4, C42.12Q8, C23.10C42, C22⋊C8.5C4, C42.6C4.12C2, (C2×C42).140C22, C2.15(M4(2)⋊4C4), C22.58(C2.C42), (C2×C4).26(C4⋊C4), (C22×C4).162(C2×C4), (C2×C4).307(C22⋊C4), SmallGroup(128,40)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.31D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=a-1b-1c3 >
Subgroups: 112 in 61 conjugacy classes, 30 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C42.6C4, C42.31D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, M4(2)⋊4C4, C42.31D4
Character table of C42.31D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | |
| 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 | -i | -i | i | -i | -i | i | -1 | i | -1 | i | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | i | -i | -1 | -1 | 1 | -i | -i | 1 | i | i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -i | -i | i | i | i | 1 | -i | 1 | i | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | -1 | 1 | i | -i | i | -1 | i | -i | -i | 1 | i | linear of order 4 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -i | i | 1 | 1 | -1 | i | -i | -1 | i | -i | i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -1 | 1 | -i | i | -i | -1 | -i | i | i | 1 | -i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | 1 | -1 | i | -i | i | 1 | -i | -i | i | -1 | -i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | 1 | -1 | -i | i | -i | 1 | i | i | -i | -1 | i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | i | -i | 1 | 1 | -1 | -i | i | -1 | -i | i | -i | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | i | i | -i | -i | -i | 1 | i | 1 | -i | -1 | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | -i | i | -1 | -1 | 1 | i | i | 1 | -i | -i | -i | linear of order 4 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | i | i | -i | -1 | -i | -1 | -i | 1 | linear of order 4 |
| 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 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2)⋊4C4 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2)⋊4C4 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2)⋊4C4 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2)⋊4C4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2)⋊4C4 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2)⋊4C4 |
►On 64 points
Generators in S64
(1 23 60 13)(2 10 61 20)(3 17 62 15)(4 12 63 22)(5 19 64 9)(6 14 57 24)(7 21 58 11)(8 16 59 18)(25 49 48 34)(26 39 41 54)(27 51 42 36)(28 33 43 56)(29 53 44 38)(30 35 45 50)(31 55 46 40)(32 37 47 52)
(1 11 64 17)(2 16 57 22)(3 13 58 19)(4 10 59 24)(5 15 60 21)(6 12 61 18)(7 9 62 23)(8 14 63 20)(25 51 44 40)(26 56 45 37)(27 53 46 34)(28 50 47 39)(29 55 48 36)(30 52 41 33)(31 49 42 38)(32 54 43 35)
(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 46 23 40 60 31 13 55)(2 43 10 56 61 28 20 33)(3 25 17 49 62 48 15 34)(4 30 12 35 63 45 22 50)(5 42 19 36 64 27 9 51)(6 47 14 52 57 32 24 37)(7 29 21 53 58 44 11 38)(8 26 16 39 59 41 18 54)
G:=sub<Sym(64)| (1,23,60,13)(2,10,61,20)(3,17,62,15)(4,12,63,22)(5,19,64,9)(6,14,57,24)(7,21,58,11)(8,16,59,18)(25,49,48,34)(26,39,41,54)(27,51,42,36)(28,33,43,56)(29,53,44,38)(30,35,45,50)(31,55,46,40)(32,37,47,52), (1,11,64,17)(2,16,57,22)(3,13,58,19)(4,10,59,24)(5,15,60,21)(6,12,61,18)(7,9,62,23)(8,14,63,20)(25,51,44,40)(26,56,45,37)(27,53,46,34)(28,50,47,39)(29,55,48,36)(30,52,41,33)(31,49,42,38)(32,54,43,35), (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), (1,46,23,40,60,31,13,55)(2,43,10,56,61,28,20,33)(3,25,17,49,62,48,15,34)(4,30,12,35,63,45,22,50)(5,42,19,36,64,27,9,51)(6,47,14,52,57,32,24,37)(7,29,21,53,58,44,11,38)(8,26,16,39,59,41,18,54)>;
G:=Group( (1,23,60,13)(2,10,61,20)(3,17,62,15)(4,12,63,22)(5,19,64,9)(6,14,57,24)(7,21,58,11)(8,16,59,18)(25,49,48,34)(26,39,41,54)(27,51,42,36)(28,33,43,56)(29,53,44,38)(30,35,45,50)(31,55,46,40)(32,37,47,52), (1,11,64,17)(2,16,57,22)(3,13,58,19)(4,10,59,24)(5,15,60,21)(6,12,61,18)(7,9,62,23)(8,14,63,20)(25,51,44,40)(26,56,45,37)(27,53,46,34)(28,50,47,39)(29,55,48,36)(30,52,41,33)(31,49,42,38)(32,54,43,35), (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), (1,46,23,40,60,31,13,55)(2,43,10,56,61,28,20,33)(3,25,17,49,62,48,15,34)(4,30,12,35,63,45,22,50)(5,42,19,36,64,27,9,51)(6,47,14,52,57,32,24,37)(7,29,21,53,58,44,11,38)(8,26,16,39,59,41,18,54) );
G=PermutationGroup([[(1,23,60,13),(2,10,61,20),(3,17,62,15),(4,12,63,22),(5,19,64,9),(6,14,57,24),(7,21,58,11),(8,16,59,18),(25,49,48,34),(26,39,41,54),(27,51,42,36),(28,33,43,56),(29,53,44,38),(30,35,45,50),(31,55,46,40),(32,37,47,52)], [(1,11,64,17),(2,16,57,22),(3,13,58,19),(4,10,59,24),(5,15,60,21),(6,12,61,18),(7,9,62,23),(8,14,63,20),(25,51,44,40),(26,56,45,37),(27,53,46,34),(28,50,47,39),(29,55,48,36),(30,52,41,33),(31,49,42,38),(32,54,43,35)], [(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)], [(1,46,23,40,60,31,13,55),(2,43,10,56,61,28,20,33),(3,25,17,49,62,48,15,34),(4,30,12,35,63,45,22,50),(5,42,19,36,64,27,9,51),(6,47,14,52,57,32,24,37),(7,29,21,53,58,44,11,38),(8,26,16,39,59,41,18,54)]])
Matrix representation of C42.31D4 ►in GL8(𝔽17)
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 4 | 0 | 0 | 0 | 0 |
| 10 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 9 | 1 | 15 |
| 0 | 0 | 0 | 0 | 8 | 8 | 1 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 9 | 7 | 10 | 0 | 0 | 0 | 0 |
| 9 | 7 | 10 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 | 14 | 16 |
| 0 | 0 | 0 | 0 | 2 | 16 | 4 | 14 |
| 0 | 0 | 0 | 0 | 6 | 15 | 10 | 16 |
| 0 | 0 | 0 | 0 | 2 | 12 | 10 | 16 |
| 7 | 7 | 3 | 3 | 0 | 0 | 0 | 0 |
| 11 | 6 | 3 | 14 | 0 | 0 | 0 | 0 |
| 4 | 5 | 10 | 6 | 0 | 0 | 0 | 0 |
| 5 | 3 | 10 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 16 | 1 | 0 |
| 0 | 0 | 0 | 0 | 14 | 10 | 6 | 1 |
| 0 | 0 | 0 | 0 | 6 | 10 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 11 | 4 | 1 |
G:=sub<GL(8,GF(17))| [0,4,0,10,0,0,0,0,13,0,7,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,6,8,0,0,0,0,15,4,9,8,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,16,10,0,0,0,0,0,1,0,0,7,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[10,7,6,9,0,0,0,0,7,7,9,7,0,0,0,0,0,0,7,10,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,9,2,6,2,0,0,0,0,4,16,15,12,0,0,0,0,14,4,10,10,0,0,0,0,16,14,16,16],[7,11,4,5,0,0,0,0,7,6,5,3,0,0,0,0,3,3,10,10,0,0,0,0,3,14,6,11,0,0,0,0,0,0,0,0,2,14,6,0,0,0,0,0,16,10,10,11,0,0,0,0,1,6,4,4,0,0,0,0,0,1,1,1] >;
C42.31D4 in GAP, Magma, Sage, TeX
C_4^2._{31}D_4 % in TeX
G:=Group("C4^2.31D4"); // GroupNames label
G:=SmallGroup(128,40);
// by ID
G=gap.SmallGroup(128,40);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,360,2804,102]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=a,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b^-1,d*c*d^-1=a^-1*b^-1*c^3>;
// generators/relations
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