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G = C23.10SD16order 128 = 27

10th non-split extension by C23 of SD16 acting via SD16/C2=D4

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

Aliases: C23.10SD16, M5(2).9C22, C8.Q86C2, (C2×C8).14D4, (C2×C8).6C23, C8.28(C4○D4), C8.17D45C2, (C2×C4).10SD16, C8.D4.3C2, C23.C8.1C2, C4.Q8.6C22, (C22×C4).107D4, C4.100(C8⋊C22), C8.C4.9C22, (C2×Q16).50C22, C22.22(C2×SD16), M4(2).C4.6C2, C4.46(C22.D4), (C2×M4(2)).38C22, C2.13(C23.46D4), (C2×C4).286(C2×D4), SmallGroup(128,971)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C23.10SD16
C1C2C4C8C2×C8C8.C4M4(2).C4 — C23.10SD16
C1C2C4C2×C8 — C23.10SD16
C1C2C2×C4C2×M4(2) — C23.10SD16
C1C2C2C2C2C4C4C2×C8 — C23.10SD16

Generators and relations for C23.10SD16
 G = < a,b,c,d,e | a2=b2=c2=1, d8=e2=c, eae-1=ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bd3 >

Subgroups: 132 in 62 conjugacy classes, 28 normal (18 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22 [×2], C8 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×4], Q8 [×2], C23, C16 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8, M4(2) [×5], Q16 [×2], C22×C4, C2×Q8, Q8⋊C4, C4.Q8, C8.C4 [×2], C8.C4, M5(2) [×2], C22⋊Q8, C2×M4(2), C2×M4(2), C2×Q16, C23.C8, C8.17D4 [×2], C8.Q8 [×2], M4(2).C4, C8.D4, C23.10SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, C4○D4 [×2], C22.D4, C2×SD16, C8⋊C22, C23.46D4, C23.10SD16

Character table of C23.10SD16

 class 12A2B2C4A4B4C4D4E8A8B8C8D8E8F8G16A16B16C16D
 size 1124224161644888888888
ρ111111111111111111111    trivial
ρ2111-111-11-111-11-1-111-11-1    linear of order 2
ρ31111111-1-111-1-1-11-11111    linear of order 2
ρ4111-111-1-11111-11-1-11-11-1    linear of order 2
ρ51111111-1-11111111-1-1-1-1    linear of order 2
ρ6111-111-1-1111-11-1-11-11-11    linear of order 2
ρ711111111111-1-1-11-1-1-1-1-1    linear of order 2
ρ8111-111-11-1111-11-1-1-11-11    linear of order 2
ρ9222-222-200-2-2000200000    orthogonal lifted from D4
ρ10222222200-2-2000-200000    orthogonal lifted from D4
ρ1122-20-220002-202i00-2i0000    complex lifted from C4○D4
ρ1222-20-22000-222i0-2i000000    complex lifted from C4○D4
ρ1322-20-220002-20-2i002i0000    complex lifted from C4○D4
ρ1422-20-22000-22-2i02i000000    complex lifted from C4○D4
ρ15222-2-2-22000000000-2--2--2-2    complex lifted from SD16
ρ162222-2-2-2000000000--2--2-2-2    complex lifted from SD16
ρ17222-2-2-22000000000--2-2-2--2    complex lifted from SD16
ρ182222-2-2-2000000000-2-2--2--2    complex lifted from SD16
ρ1944-404-400000000000000    orthogonal lifted from C8⋊C22
ρ208-8000000000000000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of C23.10SD16 in GL8(𝔽17)

016440000
101340000
134010000
13131600000
000001644
000010134
000013401
00001313160
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
0000011313
000010134
00001313160
000013401
44100000
4130160000
101340000
016440000
,
0113130000
101340000
13131600000
134010000
000013401
00004410
0000011313
000010134

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

C23.10SD16 in GAP, Magma, Sage, TeX

C_2^3._{10}{\rm SD}_{16}
% in TeX

G:=Group("C2^3.10SD16");
// GroupNames label

G:=SmallGroup(128,971);
// by ID

G=gap.SmallGroup(128,971);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,422,226,521,1684,1411,998,242,4037,1027,124]);
// Polycyclic

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

Export

Character table of C23.10SD16 in TeX

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