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## G = C8.8SD16order 128 = 27

### 8th non-split extension by C8 of SD16 acting via SD16/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C8.8SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×D4 — C8⋊6D4 — C8.8SD16
 Lower central C1 — C22 — C42 — C8.8SD16
 Upper central C1 — C22 — C42 — C8.8SD16
 Jennings C1 — C22 — C22 — C42 — C8.8SD16

Generators and relations for C8.8SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a5, cbc=a4b3 >

Subgroups: 176 in 77 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4⋊Q8, C2×M4(2), C4.6Q16, C82C8, C86D4, D42Q8, C82Q8, C8.8SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C8.C22, D4.D4, C8.D4, D4.4D4, C8.8SD16

Character table of C8.8SD16

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 8 2 2 2 2 4 8 16 16 4 4 4 4 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 0 2 -2 2 -2 -2 0 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 2 -2 2 -2 -2 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 -2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 2i -2i 0 complex lifted from C4○D4 ρ14 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 -2 2 0 -√-2 -√-2 √-2 0 0 √-2 complex lifted from SD16 ρ16 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 -2 2 0 √-2 √-2 -√-2 0 0 -√-2 complex lifted from SD16 ρ17 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 2 -2 0 -√-2 √-2 √-2 0 0 -√-2 complex lifted from SD16 ρ18 2 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 2 -2 0 √-2 -√-2 -√-2 0 0 √-2 complex lifted from SD16 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 -2√2 0 0 2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ20 4 4 -4 -4 0 0 0 0 0 0 0 0 0 2√2 0 0 -2√2 0 0 0 0 0 0 orthogonal lifted from D4.4D4 ρ21 4 -4 4 -4 0 0 -4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ22 4 -4 -4 4 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ23 4 -4 4 -4 0 0 4 0 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8.8SD16
On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 27 54 64 42 21 38 12)(2 30 55 59 43 24 39 15)(3 25 56 62 44 19 40 10)(4 28 49 57 45 22 33 13)(5 31 50 60 46 17 34 16)(6 26 51 63 47 20 35 11)(7 29 52 58 48 23 36 14)(8 32 53 61 41 18 37 9)
(2 6)(4 8)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)(25 58)(26 63)(27 60)(28 57)(29 62)(30 59)(31 64)(32 61)(33 53)(34 50)(35 55)(36 52)(37 49)(38 54)(39 51)(40 56)(41 45)(43 47)

G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,27,54,64,42,21,38,12)(2,30,55,59,43,24,39,15)(3,25,56,62,44,19,40,10)(4,28,49,57,45,22,33,13)(5,31,50,60,46,17,34,16)(6,26,51,63,47,20,35,11)(7,29,52,58,48,23,36,14)(8,32,53,61,41,18,37,9), (2,6)(4,8)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)(25,58)(26,63)(27,60)(28,57)(29,62)(30,59)(31,64)(32,61)(33,53)(34,50)(35,55)(36,52)(37,49)(38,54)(39,51)(40,56)(41,45)(43,47)>;

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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,27,54,64,42,21,38,12)(2,30,55,59,43,24,39,15)(3,25,56,62,44,19,40,10)(4,28,49,57,45,22,33,13)(5,31,50,60,46,17,34,16)(6,26,51,63,47,20,35,11)(7,29,52,58,48,23,36,14)(8,32,53,61,41,18,37,9), (2,6)(4,8)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21)(25,58)(26,63)(27,60)(28,57)(29,62)(30,59)(31,64)(32,61)(33,53)(34,50)(35,55)(36,52)(37,49)(38,54)(39,51)(40,56)(41,45)(43,47) );

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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,27,54,64,42,21,38,12),(2,30,55,59,43,24,39,15),(3,25,56,62,44,19,40,10),(4,28,49,57,45,22,33,13),(5,31,50,60,46,17,34,16),(6,26,51,63,47,20,35,11),(7,29,52,58,48,23,36,14),(8,32,53,61,41,18,37,9)], [(2,6),(4,8),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21),(25,58),(26,63),(27,60),(28,57),(29,62),(30,59),(31,64),(32,61),(33,53),(34,50),(35,55),(36,52),(37,49),(38,54),(39,51),(40,56),(41,45),(43,47)]])

Matrix representation of C8.8SD16 in GL8(𝔽17)

 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 3 2 0 0 0 0 0 14 3 0 2 0 0 0 0 0 16 14 14 0 0 0 0 1 0 3 14
,
 12 5 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 5 12 0 0 0 0 0 0 5 5 0 0 0 0 0 0 0 0 7 1 14 15 0 0 0 0 1 10 15 3 0 0 0 0 10 1 16 7 0 0 0 0 1 7 7 1
,
 1 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 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 14 14 16 0 0 0 0 0 3 14 0 16

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

C8.8SD16 in GAP, Magma, Sage, TeX

C_8._8{\rm SD}_{16}
% in TeX

G:=Group("C8.8SD16");
// GroupNames label

G:=SmallGroup(128,427);
// by ID

G=gap.SmallGroup(128,427);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,288,422,387,1123,136,2804,718,172]);
// Polycyclic

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

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