p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4:4SD16, C42.215C23, D4:C8:23C2, C4:C4.37D4, D4:Q8:5C2, C8:3Q8:14C2, (C2xD4).56D4, D4:6D4.3C2, C4.63(C4oD8), C4.37(C2xSD16), C4:Q8.35C22, D4.D4:34C2, C4.10D8:13C2, C4:C8.172C22, C4.41(C8:C22), (C4xC8).248C22, (C4xD4).42C22, C2.26(D4:D4), C2.17(C22:SD16), C22.181C22wrC2, C2.19(D4.10D4), (C2xC4).972(C2xD4), SmallGroup(128,386)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4:4SD16
G = < a,b,c,d | a4=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a-1b, dbd=a2b, dcd=c3 >
Subgroups: 288 in 120 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C42, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4xC8, D4:C4, Q8:C4, C4:C8, C4.Q8, C2.D8, C2xC4:C4, C4xD4, C4:D4, C22:Q8, C22.D4, C4:Q8, C2xSD16, C2xC4oD4, D4:C8, C4.10D8, D4.D4, D4:Q8, C8:3Q8, D4:6D4, D4:4SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2xD4, C22wrC2, C2xSD16, C4oD8, C8:C22, D4:D4, C22:SD16, D4.10D4, D4:4SD16
Character table of D4:4SD16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | √-2 | 0 | -√-2 | 0 | complex lifted from SD16 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | √-2 | 0 | -√-2 | 0 | complex lifted from SD16 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | 0 | -√2 | 0 | √2 | complex lifted from C4oD8 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | 0 | √2 | 0 | -√2 | complex lifted from C4oD8 |
| ρ19 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | 0 | √-2 | 0 | complex lifted from SD16 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | 0 | √2 | 0 | -√2 | complex lifted from C4oD8 |
| ρ21 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | 0 | √-2 | 0 | complex lifted from SD16 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | 0 | -√2 | 0 | √2 | complex lifted from C4oD8 |
| 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 | orthogonal lifted from C8:C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from D4.10D4, Schur index 2 |
►On 64 points
Generators in S64
(1 18 49 25)(2 26 50 19)(3 20 51 27)(4 28 52 21)(5 22 53 29)(6 30 54 23)(7 24 55 31)(8 32 56 17)(9 34 61 42)(10 43 62 35)(11 36 63 44)(12 45 64 37)(13 38 57 46)(14 47 58 39)(15 40 59 48)(16 41 60 33)
(1 63)(2 45)(3 57)(4 47)(5 59)(6 41)(7 61)(8 43)(9 55)(10 32)(11 49)(12 26)(13 51)(14 28)(15 53)(16 30)(17 62)(18 36)(19 64)(20 38)(21 58)(22 40)(23 60)(24 34)(25 44)(27 46)(29 48)(31 42)(33 54)(35 56)(37 50)(39 52)
(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 55)(2 50)(3 53)(4 56)(5 51)(6 54)(7 49)(8 52)(9 11)(10 14)(13 15)(17 28)(18 31)(19 26)(20 29)(21 32)(22 27)(23 30)(24 25)(34 36)(35 39)(38 40)(42 44)(43 47)(46 48)(57 59)(58 62)(61 63)
G:=sub<Sym(64)| (1,18,49,25)(2,26,50,19)(3,20,51,27)(4,28,52,21)(5,22,53,29)(6,30,54,23)(7,24,55,31)(8,32,56,17)(9,34,61,42)(10,43,62,35)(11,36,63,44)(12,45,64,37)(13,38,57,46)(14,47,58,39)(15,40,59,48)(16,41,60,33), (1,63)(2,45)(3,57)(4,47)(5,59)(6,41)(7,61)(8,43)(9,55)(10,32)(11,49)(12,26)(13,51)(14,28)(15,53)(16,30)(17,62)(18,36)(19,64)(20,38)(21,58)(22,40)(23,60)(24,34)(25,44)(27,46)(29,48)(31,42)(33,54)(35,56)(37,50)(39,52), (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), (1,55)(2,50)(3,53)(4,56)(5,51)(6,54)(7,49)(8,52)(9,11)(10,14)(13,15)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)(34,36)(35,39)(38,40)(42,44)(43,47)(46,48)(57,59)(58,62)(61,63)>;
G:=Group( (1,18,49,25)(2,26,50,19)(3,20,51,27)(4,28,52,21)(5,22,53,29)(6,30,54,23)(7,24,55,31)(8,32,56,17)(9,34,61,42)(10,43,62,35)(11,36,63,44)(12,45,64,37)(13,38,57,46)(14,47,58,39)(15,40,59,48)(16,41,60,33), (1,63)(2,45)(3,57)(4,47)(5,59)(6,41)(7,61)(8,43)(9,55)(10,32)(11,49)(12,26)(13,51)(14,28)(15,53)(16,30)(17,62)(18,36)(19,64)(20,38)(21,58)(22,40)(23,60)(24,34)(25,44)(27,46)(29,48)(31,42)(33,54)(35,56)(37,50)(39,52), (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), (1,55)(2,50)(3,53)(4,56)(5,51)(6,54)(7,49)(8,52)(9,11)(10,14)(13,15)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)(34,36)(35,39)(38,40)(42,44)(43,47)(46,48)(57,59)(58,62)(61,63) );
G=PermutationGroup([[(1,18,49,25),(2,26,50,19),(3,20,51,27),(4,28,52,21),(5,22,53,29),(6,30,54,23),(7,24,55,31),(8,32,56,17),(9,34,61,42),(10,43,62,35),(11,36,63,44),(12,45,64,37),(13,38,57,46),(14,47,58,39),(15,40,59,48),(16,41,60,33)], [(1,63),(2,45),(3,57),(4,47),(5,59),(6,41),(7,61),(8,43),(9,55),(10,32),(11,49),(12,26),(13,51),(14,28),(15,53),(16,30),(17,62),(18,36),(19,64),(20,38),(21,58),(22,40),(23,60),(24,34),(25,44),(27,46),(29,48),(31,42),(33,54),(35,56),(37,50),(39,52)], [(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)], [(1,55),(2,50),(3,53),(4,56),(5,51),(6,54),(7,49),(8,52),(9,11),(10,14),(13,15),(17,28),(18,31),(19,26),(20,29),(21,32),(22,27),(23,30),(24,25),(34,36),(35,39),(38,40),(42,44),(43,47),(46,48),(57,59),(58,62),(61,63)]])
Matrix representation of D4:4SD16 ►in GL4(F17) generated by
| 13 | 0 | 0 | 0 |
| 8 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 2 | 0 | 0 |
| 7 | 15 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 13 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 1 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [13,8,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[2,7,0,0,2,15,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,13,4,0,0,0,0,5,5,0,0,12,5],[1,15,0,0,0,16,0,0,0,0,0,1,0,0,1,0] >;
D4:4SD16 in GAP, Magma, Sage, TeX
D_4\rtimes_4{\rm SD}_{16} % in TeX
G:=Group("D4:4SD16"); // GroupNames label
G:=SmallGroup(128,386);
// by ID
G=gap.SmallGroup(128,386);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,680,422,520,1123,570,521,136,2804,1411,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^-1*b,d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations
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