p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.32D4, C42.13Q8, C23.11C42, (C2×M4(2)).4C4, C4.12(C4.D4), C4.12(C4.10D4), C4⋊M4(2).10C2, (C2×C42).141C22, C2.11(C22.C42), C22.59(C2.C42), (C2×C4).27(C4⋊C4), (C22×C4).163(C2×C4), (C2×C4).341(C22⋊C4), SmallGroup(128,41)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.32D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1b-1c3 >
Subgroups: 136 in 73 conjugacy classes, 36 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, M4(2), C22×C4, C4⋊C8, C2×C42, C2×M4(2), C4⋊M4(2), C42.32D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C4.D4, C4.10D4, C22.C42, C42.32D4
Character table of C42.32D4
| 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 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ23 | 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 C4.D4 |
| ρ24 | 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 C4.10D4, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
►On 64 points
Generators in S64
(1 23 58 9)(2 10 59 24)(3 17 60 11)(4 12 61 18)(5 19 62 13)(6 14 63 20)(7 21 64 15)(8 16 57 22)(25 53 44 37)(26 38 45 54)(27 55 46 39)(28 40 47 56)(29 49 48 33)(30 34 41 50)(31 51 42 35)(32 36 43 52)
(1 21 62 11)(2 12 63 22)(3 23 64 13)(4 14 57 24)(5 17 58 15)(6 16 59 18)(7 19 60 9)(8 10 61 20)(25 35 48 55)(26 56 41 36)(27 37 42 49)(28 50 43 38)(29 39 44 51)(30 52 45 40)(31 33 46 53)(32 54 47 34)
(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 27 23 55 58 46 9 39)(2 43 10 52 59 32 24 36)(3 25 17 53 60 44 11 37)(4 41 12 50 61 30 18 34)(5 31 19 51 62 42 13 35)(6 47 14 56 63 28 20 40)(7 29 21 49 64 48 15 33)(8 45 16 54 57 26 22 38)
G:=sub<Sym(64)| (1,23,58,9)(2,10,59,24)(3,17,60,11)(4,12,61,18)(5,19,62,13)(6,14,63,20)(7,21,64,15)(8,16,57,22)(25,53,44,37)(26,38,45,54)(27,55,46,39)(28,40,47,56)(29,49,48,33)(30,34,41,50)(31,51,42,35)(32,36,43,52), (1,21,62,11)(2,12,63,22)(3,23,64,13)(4,14,57,24)(5,17,58,15)(6,16,59,18)(7,19,60,9)(8,10,61,20)(25,35,48,55)(26,56,41,36)(27,37,42,49)(28,50,43,38)(29,39,44,51)(30,52,45,40)(31,33,46,53)(32,54,47,34), (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), (1,27,23,55,58,46,9,39)(2,43,10,52,59,32,24,36)(3,25,17,53,60,44,11,37)(4,41,12,50,61,30,18,34)(5,31,19,51,62,42,13,35)(6,47,14,56,63,28,20,40)(7,29,21,49,64,48,15,33)(8,45,16,54,57,26,22,38)>;
G:=Group( (1,23,58,9)(2,10,59,24)(3,17,60,11)(4,12,61,18)(5,19,62,13)(6,14,63,20)(7,21,64,15)(8,16,57,22)(25,53,44,37)(26,38,45,54)(27,55,46,39)(28,40,47,56)(29,49,48,33)(30,34,41,50)(31,51,42,35)(32,36,43,52), (1,21,62,11)(2,12,63,22)(3,23,64,13)(4,14,57,24)(5,17,58,15)(6,16,59,18)(7,19,60,9)(8,10,61,20)(25,35,48,55)(26,56,41,36)(27,37,42,49)(28,50,43,38)(29,39,44,51)(30,52,45,40)(31,33,46,53)(32,54,47,34), (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), (1,27,23,55,58,46,9,39)(2,43,10,52,59,32,24,36)(3,25,17,53,60,44,11,37)(4,41,12,50,61,30,18,34)(5,31,19,51,62,42,13,35)(6,47,14,56,63,28,20,40)(7,29,21,49,64,48,15,33)(8,45,16,54,57,26,22,38) );
G=PermutationGroup([[(1,23,58,9),(2,10,59,24),(3,17,60,11),(4,12,61,18),(5,19,62,13),(6,14,63,20),(7,21,64,15),(8,16,57,22),(25,53,44,37),(26,38,45,54),(27,55,46,39),(28,40,47,56),(29,49,48,33),(30,34,41,50),(31,51,42,35),(32,36,43,52)], [(1,21,62,11),(2,12,63,22),(3,23,64,13),(4,14,57,24),(5,17,58,15),(6,16,59,18),(7,19,60,9),(8,10,61,20),(25,35,48,55),(26,56,41,36),(27,37,42,49),(28,50,43,38),(29,39,44,51),(30,52,45,40),(31,33,46,53),(32,54,47,34)], [(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)], [(1,27,23,55,58,46,9,39),(2,43,10,52,59,32,24,36),(3,25,17,53,60,44,11,37),(4,41,12,50,61,30,18,34),(5,31,19,51,62,42,13,35),(6,47,14,56,63,28,20,40),(7,29,21,49,64,48,15,33),(8,45,16,54,57,26,22,38)]])
Matrix representation of C42.32D4 ►in GL8(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 1 | 0 | 0 | 0 | 0 |
| 11 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 4 | 5 | 12 | 0 | 0 | 0 | 0 |
| 4 | 5 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 5 | 5 | 7 | 7 | 0 | 0 | 0 | 0 |
| 3 | 14 | 7 | 10 | 0 | 0 | 0 | 0 |
| 15 | 6 | 12 | 14 | 0 | 0 | 0 | 0 |
| 6 | 7 | 12 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(17))| [0,1,0,11,0,0,0,0,16,0,6,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,16,10,0,0,0,0,0,1,0,0,7,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[12,5,14,4,0,0,0,0,5,5,4,5,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[5,3,15,6,0,0,0,0,5,14,6,7,0,0,0,0,7,7,12,12,0,0,0,0,7,10,14,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C42.32D4 in GAP, Magma, Sage, TeX
C_4^2._{32}D_4 % in TeX
G:=Group("C4^2.32D4"); // GroupNames label
G:=SmallGroup(128,41);
// by ID
G=gap.SmallGroup(128,41);
# by ID
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,136,2804,172]);
// 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^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^-1*b^-1*c^3>;
// generators/relations
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