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G = C23.2SD16order 128 = 27

2nd non-split extension by C23 of SD16 acting via SD16/C2=D4

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C23.2SD16, C4.8C4≀C2, (C2×C4).2D8, (C2×C8).22D4, C4.Q8.2C4, (C2×Q16).4C4, C4.4(C23⋊C4), C8.D4.2C2, C23.C8.5C2, (C22×C4).30D4, C4.10C42.1C2, C2.7(C22.SD16), (C2×M4(2)).4C22, C22.16(D4⋊C4), (C2×C8).4(C2×C4), (C2×C4).55(C22⋊C4), SmallGroup(128,74)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C23.2SD16
C1C2C4C2×C4C22×C4C2×M4(2)C8.D4 — C23.2SD16
C1C2C2×C4C2×C8 — C23.2SD16
C1C2C2×C4C2×M4(2) — C23.2SD16
C1C2C2C2C2C4C2×C4C2×M4(2) — C23.2SD16

Generators and relations for C23.2SD16
 G = < a,b,c,d,e | a2=b2=c2=1, d8=e2=c, dad-1=eae-1=ab=ba, ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=abcd3 >

2C2
4C2
2C22
2C4
4C22
8C4
8C4
2C8
2C2×C4
2C2×C4
2C8
4C8
4C2×C4
4Q8
4Q8
4C2×C4
4C8
2C2×C8
2C2×Q8
2C4⋊C4
2C2×C8
4M4(2)
4C16
4Q16
4M4(2)
4C4⋊C4
4C22⋊C4
4M4(2)
2M5(2)
2C2×M4(2)
2Q8⋊C4
2C22⋊Q8

Character table of C23.2SD16

 class 12A2B2C4A4B4C4D4E8A8B8C8D8E8F8G16A16B16C16D
 size 1124224161644888888888
ρ111111111111111111111    trivial
ρ21111111-1-111-1-1-11-11111    linear of order 2
ρ31111111-1-11111111-1-1-1-1    linear of order 2
ρ411111111111-1-1-11-1-1-1-1-1    linear of order 2
ρ5111-111-11-1-1-1ii-i1-iii-i-i    linear of order 4
ρ6111-111-11-1-1-1-i-ii1i-i-iii    linear of order 4
ρ7111-111-1-11-1-1ii-i1-i-i-iii    linear of order 4
ρ8111-111-1-11-1-1-i-ii1iii-i-i    linear of order 4
ρ9222-222-20022000-200000    orthogonal lifted from D4
ρ10222222200-2-2000-200000    orthogonal lifted from D4
ρ11222-2-2-220000000002-22-2    orthogonal lifted from D8
ρ12222-2-2-22000000000-22-22    orthogonal lifted from D8
ρ1322-202-20002i-2i-1-i1+i-1+i01-i0000    complex lifted from C4≀C2
ρ1422-202-2000-2i2i1-i-1+i1+i0-1-i0000    complex lifted from C4≀C2
ρ152222-2-2-2000000000-2--2--2-2    complex lifted from SD16
ρ162222-2-2-2000000000--2-2-2--2    complex lifted from SD16
ρ1722-202-2000-2i2i-1+i1-i-1-i01+i0000    complex lifted from C4≀C2
ρ1822-202-20002i-2i1+i-1-i1-i0-1+i0000    complex lifted from C4≀C2
ρ1944-40-4400000000000000    orthogonal lifted from C23⋊C4
ρ208-8000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C23.2SD16
On 32 points
Generators in S32
(1 30)(2 23)(3 24)(4 17)(5 18)(6 27)(7 28)(8 21)(9 22)(10 31)(11 32)(12 25)(13 26)(14 19)(15 20)(16 29)
(1 9)(3 11)(5 13)(7 15)(18 26)(20 28)(22 30)(24 32)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 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 18 9 26)(2 8 10 16)(3 32 11 24)(4 14 12 6)(5 30 13 22)(7 28 15 20)(17 19 25 27)(21 31 29 23)

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

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

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

Matrix representation of C23.2SD16 in GL8(𝔽17)

100150000
016200000
00100000
000160000
15014200016
09810010
09810100
15014216000
,
160000000
016000000
001600000
000160000
2713131000
149440100
149440010
2713130001
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
149440020
15104400015
000010016
000001610
21513221505
11688152111
81691152111
216121221505
,
016200000
100150000
000160000
00100000
1381116143413
1316111331313
156147413314
0130513131414

G:=sub<GL(8,GF(17))| [1,0,0,0,15,0,0,15,0,16,0,0,0,9,9,0,0,2,1,0,14,8,8,14,15,0,0,16,2,1,1,2,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0],[16,0,0,0,2,14,14,2,0,16,0,0,7,9,9,7,0,0,16,0,13,4,4,13,0,0,0,16,13,4,4,13,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[14,15,0,0,2,1,8,2,9,10,0,0,15,16,16,16,4,4,0,0,13,8,9,12,4,4,0,0,2,8,1,12,0,0,1,0,2,15,15,2,0,0,0,16,15,2,2,15,2,0,0,1,0,1,1,0,0,15,16,0,5,11,11,5],[0,1,0,0,13,13,15,0,16,0,0,0,8,16,6,13,2,0,0,1,11,1,14,0,0,15,16,0,16,11,7,5,0,0,0,0,14,3,4,13,0,0,0,0,3,3,13,13,0,0,0,0,4,13,3,14,0,0,0,0,13,13,14,14] >;

C23.2SD16 in GAP, Magma, Sage, TeX

C_2^3._2{\rm SD}_{16}
% in TeX

G:=Group("C2^3.2SD16");
// GroupNames label

G:=SmallGroup(128,74);
// by ID

G=gap.SmallGroup(128,74);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,520,1690,521,248,1411,172,4037,2028,124]);
// Polycyclic

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

Export

Subgroup lattice of C23.2SD16 in TeX
Character table of C23.2SD16 in TeX

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