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G = SD162D4order 128 = 27

2nd semidirect product of SD16 and D4 acting via D4/C4=C2

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

Aliases: SD162D4, C42.442C23, C4.1292+ 1+4, C2.59D42, C8.7(C2×D4), C86D46C2, D45D45C2, C82D422C2, C4⋊C831C22, C4⋊C4.151D4, (C4×C8)⋊32C22, Q85D44C2, D4.24(C2×D4), Q8.22(C2×D4), D4⋊D439C2, Q8⋊D418C2, (C4×SD16)⋊20C2, (C2×D4).165D4, C8.D421C2, (C2×C8).90C23, (C4×Q8)⋊21C22, C4.89(C22×D4), C4.Q853C22, D4.2D439C2, C22⋊SD1619C2, D4.7D439C2, C8.12D422C2, C4⋊C4.214C23, C4⋊D411C22, C22⋊C827C22, (C2×C4).473C24, Q8.D438C2, C22⋊C4.161D4, (C2×Q16)⋊29C22, (C2×D8).81C22, C23.315(C2×D4), D4⋊C439C22, C2.61(D4○SD16), Q8⋊C484C22, (C2×SD16)⋊47C22, (C2×D4).212C23, (C4×D4).147C22, C4.4D414C22, (C2×Q8).196C23, (C22×Q8)⋊24C22, C22⋊Q8.60C22, (C2×M4(2))⋊25C22, (C22×C4).323C23, C22.733(C22×D4), C2.80(D8⋊C22), (C22×D4).401C22, (C2×C8⋊C22)⋊33C2, (C2×C4).157(C2×D4), (C2×C8.C22)⋊30C2, (C2×C4○D4).188C22, SmallGroup(128,2007)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — SD162D4
C1C2C22C2×C4C2×D4C22×D4C2×C8⋊C22 — SD162D4
C1C2C2×C4 — SD162D4
C1C22C4×D4 — SD162D4
C1C2C2C2×C4 — SD162D4

Generators and relations for SD162D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, dad=a5, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 544 in 249 conjugacy classes, 94 normal (84 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×20], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×16], D4 [×2], D4 [×15], Q8 [×2], Q8 [×7], C23 [×2], C23 [×8], C42, C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×4], M4(2) [×4], D8 [×4], SD16 [×4], SD16 [×6], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×6], C24, C4×C8, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C2×C22⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2, C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8, C22.D4, C4.4D4 [×2], C4.4D4, C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×5], C2×Q16 [×2], C8⋊C22 [×4], C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4 [×2], C86D4, C4×SD16, Q8⋊D4, D4⋊D4, C22⋊SD16, D4.7D4, D4.2D4, Q8.D4, C82D4, C8.D4, C8.12D4, D45D4, Q85D4, C2×C8⋊C22, C2×C8.C22, SD162D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D8⋊C22, D4○SD16, SD162D4

Character table of SD162D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114444882222444448888444488
ρ111111111111111111111111111111    trivial
ρ21111111-11-1-1-1111-1-1-1-1-11-11-1-111-11    linear of order 2
ρ311111-1-1-1-1-1-1-1111-1-1-1-1111111-1-11-1    linear of order 2
ρ411111-1-11-11111111111-11-11-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-111-1-1-111-11-1111-1-11-1-1111-1    linear of order 2
ρ61111-1-1-1-1111111-1-11-1-1-1-1111111-1-1    linear of order 2
ρ71111-111-1-111111-1-11-1-11-1-11-1-1-1-111    linear of order 2
ρ81111-1111-1-1-1-111-11-111-1-11111-1-1-11    linear of order 2
ρ91111-111-1-1-11111-111-1111-1-11111-1-1    linear of order 2
ρ101111-1111-11-1-111-1-1-11-1-111-1-1-1111-1    linear of order 2
ρ111111-1-1-1111-1-111-1-1-11-111-1-111-1-1-11    linear of order 2
ρ121111-1-1-1-11-11111-111-11-111-1-1-1-1-111    linear of order 2
ρ1311111-1-1-1-11-1-11111-1-111-11-1-1-111-11    linear of order 2
ρ1411111-1-11-1-111111-111-1-1-1-1-1111111    linear of order 2
ρ15111111111-111111-111-11-11-1-1-1-1-1-1-1    linear of order 2
ρ161111111-111-1-11111-1-11-1-1-1-111-1-11-1    linear of order 2
ρ172222-200-200-2-2-2-2202200000000000    orthogonal lifted from D4
ρ182222200-20022-2-2-20-2200000000000    orthogonal lifted from D4
ρ192-22-202-2000002-20200-200002-20000    orthogonal lifted from D4
ρ202222-20020022-2-220-2-200000000000    orthogonal lifted from D4
ρ212-22-202-2000002-20-20020000-220000    orthogonal lifted from D4
ρ222222200200-2-2-2-2-202-200000000000    orthogonal lifted from D4
ρ232-22-20-22000002-20-200200002-20000    orthogonal lifted from D4
ρ242-22-20-22000002-20200-20000-220000    orthogonal lifted from D4
ρ254-44-400000000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-440000004i-4i00000000000000000    complex lifted from D8⋊C22
ρ274-4-44000000-4i4i00000000000000000    complex lifted from D8⋊C22
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of SD162D4
On 32 points
Generators in S32
(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)
(1 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 10 23 32)(2 11 24 25)(3 12 17 26)(4 13 18 27)(5 14 19 28)(6 15 20 29)(7 16 21 30)(8 9 22 31)
(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)

G:=sub<Sym(32)| (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), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,10,23,32)(2,11,24,25)(3,12,17,26)(4,13,18,27)(5,14,19,28)(6,15,20,29)(7,16,21,30)(8,9,22,31), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)>;

G:=Group( (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), (1,23)(2,18)(3,21)(4,24)(5,19)(6,22)(7,17)(8,20)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,10,23,32)(2,11,24,25)(3,12,17,26)(4,13,18,27)(5,14,19,28)(6,15,20,29)(7,16,21,30)(8,9,22,31), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21) );

G=PermutationGroup([(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)], [(1,23),(2,18),(3,21),(4,24),(5,19),(6,22),(7,17),(8,20),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30)], [(1,10,23,32),(2,11,24,25),(3,12,17,26),(4,13,18,27),(5,14,19,28),(6,15,20,29),(7,16,21,30),(8,9,22,31)], [(1,32),(2,29),(3,26),(4,31),(5,28),(6,25),(7,30),(8,27),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)])

Matrix representation of SD162D4 in GL6(𝔽17)

1600000
0160000
000010
00161612
000100
00161611
,
100000
010000
001000
0001600
00161612
0010016
,
0160000
100000
0001300
004000
0044139
0001344
,
0160000
1600000
0001300
004000
00131348
00401313

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,16,0,0,0,16,1,16,0,0,1,1,0,1,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,1,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,2,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,4,4,0,0,0,13,0,4,13,0,0,0,0,13,4,0,0,0,0,9,4],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,4,13,4,0,0,13,0,13,0,0,0,0,0,4,13,0,0,0,0,8,13] >;

SD162D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_2D_4
% in TeX

G:=Group("SD16:2D4");
// GroupNames label

G:=SmallGroup(128,2007);
// by ID

G=gap.SmallGroup(128,2007);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,346,2804,1411,375,172]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^3,a*c=c*a,d*a*d=a^5,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of SD162D4 in TeX

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