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G = C23⋊2SD16order 128 = 27

2nd semidirect product of C23 and SD16 acting via SD16/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C23⋊2SD16
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23⋊3D4 — C23⋊2SD16
 Lower central C1 — C22 — C22×C4 — C23⋊2SD16
 Upper central C1 — C22 — C22×C4 — C23⋊2SD16
 Jennings C1 — C2 — C22 — C22×C4 — C23⋊2SD16

Generators and relations for C232SD16
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d3 >

Subgroups: 452 in 154 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C24, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C2×SD16, C22×D4, C22×Q8, C23⋊C8, C22.SD16, C23⋊Q8, Q8⋊D4, C233D4, C232SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8⋊C22, C22⋊SD16, D4.9D4, C2≀C22, C232SD16

Character table of C232SD16

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 8 8 4 4 8 8 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 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 -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 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 -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 -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 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 -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 1 -1 1 linear of order 2 ρ9 2 2 2 2 -2 -2 0 0 0 2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 0 0 -2 0 -2 2 0 0 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 -2 -2 0 0 -2 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -2 0 0 0 -2 2 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 0 0 2 0 -2 2 0 0 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 2 2 2 2 0 0 -2 -2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 -2 2 2 -2 0 0 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 2 0 0 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 -2 0 0 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 2 0 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ19 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 2 0 -2 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ20 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 0 0 0 orthogonal lifted from C2≀C22 ρ21 4 4 -4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 -2i 0 0 0 0 0 2i 0 0 0 0 complex lifted from D4.9D4 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 0 2i 0 0 0 0 0 -2i 0 0 0 0 complex lifted from D4.9D4

Smallest permutation representation of C232SD16
On 32 points
Generators in S32
(2 29)(3 14)(4 20)(6 25)(7 10)(8 24)(9 22)(11 27)(13 18)(15 31)(19 30)(23 26)
(1 28)(2 18)(3 30)(4 20)(5 32)(6 22)(7 26)(8 24)(9 25)(10 23)(11 27)(12 17)(13 29)(14 19)(15 31)(16 21)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 28)(18 31)(19 26)(20 29)(21 32)(22 27)(23 30)(24 25)

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

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

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

Matrix representation of C232SD16 in GL6(𝔽17)

 16 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 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 5 12 0 0 0 0 5 5 0 0 0 0 0 0 9 9 9 8 0 0 8 8 9 8 0 0 9 8 9 9 0 0 9 8 8 8
,
 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 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,9,8,9,9,0,0,9,8,8,8,0,0,9,9,9,8,0,0,8,8,9,8],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C232SD16 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2{\rm SD}_{16}
% in TeX

G:=Group("C2^3:2SD16");
// GroupNames label

G:=SmallGroup(128,333);
// by ID

G=gap.SmallGroup(128,333);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,1123,570,521,136,1411]);
// Polycyclic

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

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