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G = D4.5SD16  order 128 = 27

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

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

Aliases: D4.5SD16, C42.204C23, Q8⋊C8⋊26C2, C4⋊C4.56D4, D4⋊C8.10C2, (C2×Q8).50D4, Q8⋊Q8⋊32C2, (C2×D4).255D4, C4.36(C4○D8), (C4×C8).25C22, D4⋊3Q8.2C2, C4.SD16⋊4C2, C4.33(C2×SD16), C4⋊Q8.24C22, C4⋊C8.167C22, C4.6Q16⋊13C2, D4.D4.3C2, (C4×D4).34C22, (C4×Q8).34C22, C2.14(Q8⋊D4), C4.65(C8.C22), C22.170C22≀C2, C2.16(D4.9D4), C2.21(D4.7D4), (C2×C4).961(C2×D4), SmallGroup(128,375)

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, D4⋊3Q8, 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 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 111144222244488881644448888
ρ111111111111111111111111111    trivial
ρ21111-1-111111111-1-1-11-1-1-1-1-11-11    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-111111111-1-1-1-111111-11-1    linear of order 2
ρ51111-1-11111-1-11-1-111-1-1-1-1-11111    linear of order 2
ρ61111111111-1-11-11-1-1-11111-11-11    linear of order 2
ρ71111-1-11111-1-11-1-11111111-1-1-1-1    linear of order 2
ρ81111111111-1-11-11-1-11-1-1-1-11-11-1    linear of order 2
ρ9222222-2-22200-20-200000000000    orthogonal lifted from D4
ρ10222200-2-2-2-2002002-2000000000    orthogonal lifted from D4
ρ11222200-2-2-2-200200-22000000000    orthogonal lifted from D4
ρ1222220022-2-2-2-2-22000000000000    orthogonal lifted from D4
ρ1322220022-2-222-2-2000000000000    orthogonal lifted from D4
ρ142222-2-2-2-22200-20200000000000    orthogonal lifted from D4
ρ152-22-2-2200-2200000000-√-2-√-2√-2√-20-√-20√-2    complex lifted from SD16
ρ162-22-22-200-2200000000√-2√-2-√-2-√-20-√-20√-2    complex lifted from SD16
ρ172-22-2-2200-2200000000√-2√-2-√-2-√-20√-20-√-2    complex lifted from SD16
ρ182-22-22-200-2200000000-√-2-√-2√-2√-20√-20-√-2    complex lifted from SD16
ρ192-2-22002-200-2i2i000000-√2√2√2-√2√-20-√-20    complex lifted from C4○D8
ρ202-2-22002-200-2i2i000000√2-√2-√2√2-√-20√-20    complex lifted from C4○D8
ρ212-2-22002-2002i-2i000000-√2√2√2-√2-√-20√-20    complex lifted from C4○D8
ρ222-2-22002-2002i-2i000000√2-√2-√2√2√-20-√-20    complex lifted from C4○D8
ρ234-4-4400-44000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ244-44-400004-40000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2544-4-400000000000000-2i2i-2i2i0000    complex lifted from D4.9D4
ρ2644-4-4000000000000002i-2i2i-2i0000    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

0100
16000
0010
0001
,
0100
1000
00160
00016
,
3300
14300
00150
0009
,
51200
121200
0002
0090
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
 

Export

Character table of D4.5SD16 in TeX

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