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## G = D4.5SD16order 128 = 27

### 1st non-split extension by D4 of SD16 acting via SD16/Q8=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — D4.5SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — D4⋊3Q8 — D4.5SD16
 Lower central C1 — C22 — C42 — D4.5SD16
 Upper central C1 — C22 — C42 — D4.5SD16
 Jennings C1 — C22 — C22 — C42 — D4.5SD16

Generators and relations for D4.5SD16
G = < a,b,c,d | a4=b2=c8=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=dbd-1=ab, dcd-1=a2c3 >

Subgroups: 232 in 104 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×C8, Q8⋊C4, C4⋊C8, C4.Q8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×SD16, D4⋊C8, Q8⋊C8, C4.6Q16, D4.D4, Q8⋊Q8, C4.SD16, D43Q8, D4.5SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C4○D8, C8.C22, Q8⋊D4, D4.7D4, D4.9D4, D4.5SD16

Character table of D4.5SD16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 4 4 2 2 2 2 4 4 4 8 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 2 2 -2 -2 2 2 0 0 -2 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 -2 -2 0 0 2 0 0 2 -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 0 0 2 0 0 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 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 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 -2 -2 2 2 0 0 -2 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 2 -2 -2 2 0 0 -2 2 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 0 -√-2 0 √-2 complex lifted from SD16 ρ16 2 -2 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 0 -√-2 0 √-2 complex lifted from SD16 ρ17 2 -2 2 -2 -2 2 0 0 -2 2 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 0 √-2 0 -√-2 complex lifted from SD16 ρ18 2 -2 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 0 √-2 0 -√-2 complex lifted from SD16 ρ19 2 -2 -2 2 0 0 2 -2 0 0 -2i 2i 0 0 0 0 0 0 -√2 √2 √2 -√2 √-2 0 -√-2 0 complex lifted from C4○D8 ρ20 2 -2 -2 2 0 0 2 -2 0 0 -2i 2i 0 0 0 0 0 0 √2 -√2 -√2 √2 -√-2 0 √-2 0 complex lifted from C4○D8 ρ21 2 -2 -2 2 0 0 2 -2 0 0 2i -2i 0 0 0 0 0 0 -√2 √2 √2 -√2 -√-2 0 √-2 0 complex lifted from C4○D8 ρ22 2 -2 -2 2 0 0 2 -2 0 0 2i -2i 0 0 0 0 0 0 √2 -√2 -√2 √2 √-2 0 -√-2 0 complex lifted from C4○D8 ρ23 4 -4 -4 4 0 0 -4 4 0 0 0 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 0 0 0 4 -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 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i -2i 2i 0 0 0 0 complex lifted from D4.9D4 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i 2i -2i 0 0 0 0 complex lifted from D4.9D4

Smallest permutation representation of D4.5SD16
On 64 points
Generators in S64
(1 49 27 9)(2 50 28 10)(3 51 29 11)(4 52 30 12)(5 53 31 13)(6 54 32 14)(7 55 25 15)(8 56 26 16)(17 59 40 46)(18 60 33 47)(19 61 34 48)(20 62 35 41)(21 63 36 42)(22 64 37 43)(23 57 38 44)(24 58 39 45)
(1 13)(2 32)(3 55)(4 8)(5 9)(6 28)(7 51)(10 14)(11 25)(12 56)(15 29)(16 52)(17 21)(18 43)(19 38)(20 58)(22 47)(23 34)(24 62)(26 30)(27 53)(31 49)(33 64)(35 45)(36 40)(37 60)(39 41)(42 59)(44 48)(46 63)(50 54)(57 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 38 27 23)(2 18 28 33)(3 36 29 21)(4 24 30 39)(5 34 31 19)(6 22 32 37)(7 40 25 17)(8 20 26 35)(9 44 49 57)(10 60 50 47)(11 42 51 63)(12 58 52 45)(13 48 53 61)(14 64 54 43)(15 46 55 59)(16 62 56 41)

G:=sub<Sym(64)| (1,49,27,9)(2,50,28,10)(3,51,29,11)(4,52,30,12)(5,53,31,13)(6,54,32,14)(7,55,25,15)(8,56,26,16)(17,59,40,46)(18,60,33,47)(19,61,34,48)(20,62,35,41)(21,63,36,42)(22,64,37,43)(23,57,38,44)(24,58,39,45), (1,13)(2,32)(3,55)(4,8)(5,9)(6,28)(7,51)(10,14)(11,25)(12,56)(15,29)(16,52)(17,21)(18,43)(19,38)(20,58)(22,47)(23,34)(24,62)(26,30)(27,53)(31,49)(33,64)(35,45)(36,40)(37,60)(39,41)(42,59)(44,48)(46,63)(50,54)(57,61), (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), (1,38,27,23)(2,18,28,33)(3,36,29,21)(4,24,30,39)(5,34,31,19)(6,22,32,37)(7,40,25,17)(8,20,26,35)(9,44,49,57)(10,60,50,47)(11,42,51,63)(12,58,52,45)(13,48,53,61)(14,64,54,43)(15,46,55,59)(16,62,56,41)>;

G:=Group( (1,49,27,9)(2,50,28,10)(3,51,29,11)(4,52,30,12)(5,53,31,13)(6,54,32,14)(7,55,25,15)(8,56,26,16)(17,59,40,46)(18,60,33,47)(19,61,34,48)(20,62,35,41)(21,63,36,42)(22,64,37,43)(23,57,38,44)(24,58,39,45), (1,13)(2,32)(3,55)(4,8)(5,9)(6,28)(7,51)(10,14)(11,25)(12,56)(15,29)(16,52)(17,21)(18,43)(19,38)(20,58)(22,47)(23,34)(24,62)(26,30)(27,53)(31,49)(33,64)(35,45)(36,40)(37,60)(39,41)(42,59)(44,48)(46,63)(50,54)(57,61), (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), (1,38,27,23)(2,18,28,33)(3,36,29,21)(4,24,30,39)(5,34,31,19)(6,22,32,37)(7,40,25,17)(8,20,26,35)(9,44,49,57)(10,60,50,47)(11,42,51,63)(12,58,52,45)(13,48,53,61)(14,64,54,43)(15,46,55,59)(16,62,56,41) );

G=PermutationGroup([[(1,49,27,9),(2,50,28,10),(3,51,29,11),(4,52,30,12),(5,53,31,13),(6,54,32,14),(7,55,25,15),(8,56,26,16),(17,59,40,46),(18,60,33,47),(19,61,34,48),(20,62,35,41),(21,63,36,42),(22,64,37,43),(23,57,38,44),(24,58,39,45)], [(1,13),(2,32),(3,55),(4,8),(5,9),(6,28),(7,51),(10,14),(11,25),(12,56),(15,29),(16,52),(17,21),(18,43),(19,38),(20,58),(22,47),(23,34),(24,62),(26,30),(27,53),(31,49),(33,64),(35,45),(36,40),(37,60),(39,41),(42,59),(44,48),(46,63),(50,54),(57,61)], [(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)], [(1,38,27,23),(2,18,28,33),(3,36,29,21),(4,24,30,39),(5,34,31,19),(6,22,32,37),(7,40,25,17),(8,20,26,35),(9,44,49,57),(10,60,50,47),(11,42,51,63),(12,58,52,45),(13,48,53,61),(14,64,54,43),(15,46,55,59),(16,62,56,41)]])

Matrix representation of D4.5SD16 in GL4(𝔽17) generated by

 0 1 0 0 16 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 16 0 0 0 0 16
,
 3 3 0 0 14 3 0 0 0 0 15 0 0 0 0 9
,
 5 12 0 0 12 12 0 0 0 0 0 2 0 0 9 0
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[3,14,0,0,3,3,0,0,0,0,15,0,0,0,0,9],[5,12,0,0,12,12,0,0,0,0,0,9,0,0,2,0] >;

D4.5SD16 in GAP, Magma, Sage, TeX

D_4._5{\rm SD}_{16}
% in TeX

G:=Group("D4.5SD16");
// GroupNames label

G:=SmallGroup(128,375);
// by ID

G=gap.SmallGroup(128,375);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,232,422,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a*b,d*c*d^-1=a^2*c^3>;
// generators/relations

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