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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C23.10SD16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C8.C4 — M4(2).C4 — C23.10SD16
 Lower central C1 — C2 — C4 — C2×C8 — C23.10SD16
 Upper central C1 — C2 — C2×C4 — C2×M4(2) — C23.10SD16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×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 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 8C 8D 8E 8F 8G 16A 16B 16C 16D size 1 1 2 4 2 2 4 16 16 4 4 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 1 1 linear of order 2 ρ4 1 1 1 -1 1 1 -1 -1 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 linear of order 2 ρ8 1 1 1 -1 1 1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ9 2 2 2 -2 2 2 -2 0 0 -2 -2 0 0 0 2 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 2 0 0 -2 -2 0 0 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 0 -2 2 0 0 0 2 -2 0 2i 0 0 -2i 0 0 0 0 complex lifted from C4○D4 ρ12 2 2 -2 0 -2 2 0 0 0 -2 2 2i 0 -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ13 2 2 -2 0 -2 2 0 0 0 2 -2 0 -2i 0 0 2i 0 0 0 0 complex lifted from C4○D4 ρ14 2 2 -2 0 -2 2 0 0 0 -2 2 -2i 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ16 2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ17 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ18 2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 complex lifted from SD16 ρ19 4 4 -4 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ20 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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)

 0 16 4 4 0 0 0 0 1 0 13 4 0 0 0 0 13 4 0 1 0 0 0 0 13 13 16 0 0 0 0 0 0 0 0 0 0 16 4 4 0 0 0 0 1 0 13 4 0 0 0 0 13 4 0 1 0 0 0 0 13 13 16 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 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 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 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

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

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