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G = D4.1Q16order 128 = 27

1st non-split extension by D4 of Q16 acting via Q16/C8=C2

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

Aliases: D4.1Q16, C8.26SD16, C42.226C23, C81C87C2, C82Q85C2, (C8×D4).6C2, C4⋊C4.201D4, (C2×C8).345D4, C4.25(C2×Q16), (C2×D4).191D4, C4.42(C4○D8), (C4×C8).55C22, D42Q8.8C2, C4.59(C2×SD16), C4⋊Q8.49C22, C4⋊C8.178C22, C4.6Q1615C2, C2.8(D4.D4), C2.8(D4.4D4), C2.8(C8.18D4), (C4×D4).279C22, C4.114(C8.C22), C22.187(C4⋊D4), (C2×C4).11(C4○D4), (C2×C4).1261(C2×D4), SmallGroup(128,407)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — D4.1Q16
C1C2C22C2×C4C42C4×D4C8×D4 — D4.1Q16
C1C22C42 — D4.1Q16
C1C22C42 — D4.1Q16
C1C22C22C42 — D4.1Q16

Generators and relations for D4.1Q16
 G = < a,b,c,d | a4=b2=c8=1, d2=a2c4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

Subgroups: 176 in 80 conjugacy classes, 36 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×4], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], C22×C4, C2×D4, C2×Q8 [×2], C4×C8, C22⋊C8, D4⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C4×D4, C4⋊Q8 [×2], C22×C8, C4.6Q16 [×2], C81C8, C8×D4, D42Q8 [×2], C82Q8, D4.1Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], Q16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C2×Q16, C4○D8, C8.C22, D4.D4, C8.18D4, D4.4D4, D4.1Q16

Character table of D4.1Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111442222444161622224444448888
ρ111111111111111111111111111111    trivial
ρ21111-1-11111-11-11-1-1-1-1-1111-1-111-11-1    linear of order 2
ρ31111-1-11111-11-1-1-11111-1-1-111-11111    linear of order 2
ρ41111111111111-11-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ511111111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ61111-1-11111-11-1111111-1-1-111-1-1-1-1-1    linear of order 2
ρ71111-1-11111-11-1-11-1-1-1-1111-1-11-11-11    linear of order 2
ρ81111111111111-1-11111111111-1-1-1-1    linear of order 2
ρ92222-2-2-22-222-220000000000000000    orthogonal lifted from D4
ρ102222002-22-20-20002222000-2-200000    orthogonal lifted from D4
ρ112222002-22-20-2000-2-2-2-20002200000    orthogonal lifted from D4
ρ12222222-22-22-2-2-20000000000000000    orthogonal lifted from D4
ρ1322-2-22-2020-2000002-2-22-222-22-20000    symplectic lifted from Q16, Schur index 2
ρ1422-2-2-22020-2000002-2-222-2-2-2220000    symplectic lifted from Q16, Schur index 2
ρ1522-2-22-2020-200000-222-22-2-22-220000    symplectic lifted from Q16, Schur index 2
ρ1622-2-2-22020-200000-222-2-2222-2-20000    symplectic lifted from Q16, Schur index 2
ρ17222200-2-2-2-2020000000-2i-2i2i002i0000    complex lifted from C4○D4
ρ1822-2-2000-202-2i02i002-2-22--2-2--22-2-20000    complex lifted from C4○D8
ρ192-22-200-2020000002-22-2000000-2-2--2--2    complex lifted from SD16
ρ202-22-200-202000000-22-22000000--2-2-2--2    complex lifted from SD16
ρ212-22-200-2020000002-22-2000000--2--2-2-2    complex lifted from SD16
ρ22222200-2-2-2-20200000002i2i-2i00-2i0000    complex lifted from C4○D4
ρ2322-2-2000-2022i0-2i00-222-2--2-2--2-22-20000    complex lifted from C4○D8
ρ2422-2-2000-2022i0-2i002-2-22-2--2-22-2--20000    complex lifted from C4○D8
ρ252-22-200-202000000-22-22000000-2--2--2-2    complex lifted from SD16
ρ2622-2-2000-202-2i02i00-222-2-2--2-2-22--20000    complex lifted from C4○D8
ρ274-4-44000000000002222-22-220000000000    orthogonal lifted from D4.4D4
ρ284-4-4400000000000-22-2222220000000000    orthogonal lifted from D4.4D4
ρ294-44-40040-400000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of D4.1Q16
On 64 points
Generators in S64
(1 35 43 53)(2 36 44 54)(3 37 45 55)(4 38 46 56)(5 39 47 49)(6 40 48 50)(7 33 41 51)(8 34 42 52)(9 19 63 31)(10 20 64 32)(11 21 57 25)(12 22 58 26)(13 23 59 27)(14 24 60 28)(15 17 61 29)(16 18 62 30)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(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 58 47 16)(2 57 48 15)(3 64 41 14)(4 63 42 13)(5 62 43 12)(6 61 44 11)(7 60 45 10)(8 59 46 9)(17 54 25 40)(18 53 26 39)(19 52 27 38)(20 51 28 37)(21 50 29 36)(22 49 30 35)(23 56 31 34)(24 55 32 33)

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

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

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

Matrix representation of D4.1Q16 in GL4(𝔽17) generated by

11500
11600
0010
0001
,
11500
01600
00160
00016
,
16000
01600
00314
0033
,
0700
12000
0071
00110
G:=sub<GL(4,GF(17))| [1,1,0,0,15,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,15,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,3,3,0,0,14,3],[0,12,0,0,7,0,0,0,0,0,7,1,0,0,1,10] >;

D4.1Q16 in GAP, Magma, Sage, TeX

D_4._1Q_{16}
% in TeX

G:=Group("D4.1Q16");
// GroupNames label

G:=SmallGroup(128,407);
// by ID

G=gap.SmallGroup(128,407);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,288,422,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of D4.1Q16 in TeX

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