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## G = C8.13SD16order 128 = 27

### 13rd non-split extension by C8 of SD16 acting via SD16/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C8.13SD16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C16⋊5C4 — C8.13SD16
 Lower central C1 — C2 — C4 — C2×C8 — C8.13SD16
 Upper central C1 — C22 — C42 — C4×C8 — C8.13SD16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.13SD16

Generators and relations for C8.13SD16
G = < a,b,c | a8=c2=1, b8=a4, bab-1=a5, cac=a-1, cbc=a6b3 >

Subgroups: 248 in 76 conjugacy classes, 32 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], C23 [×2], C16 [×2], C42, C4⋊C4 [×4], C2×C8 [×2], D8 [×6], C2×D4 [×4], C4×C8, C4.Q8, C2.D8 [×2], C2×C16 [×2], C42.C2, C41D4, C2×D8 [×2], C2×D8, C165C4, C2.D16 [×4], C84D4, C8.5Q8, C8.13SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], C4.4D4, C2×D8, C2×SD16, C4.4D8, C16⋊C22 [×2], C8.13SD16

Character table of C8.13SD16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 16 16 2 2 4 4 16 16 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 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 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 -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 -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 -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 -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 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 1 1 1 linear of order 2 ρ9 2 2 2 2 0 0 2 2 -2 -2 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 2 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 0 √2 √2 -√2 √2 -√2 -√2 -√2 √2 orthogonal lifted from D8 ρ12 2 2 2 2 0 0 -2 -2 2 -2 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ13 2 2 2 2 0 0 -2 -2 2 -2 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ14 2 2 2 2 0 0 -2 -2 -2 2 0 0 0 0 0 0 0 0 -√2 -√2 √2 -√2 √2 √2 √2 -√2 orthogonal lifted from D8 ρ15 2 -2 2 -2 0 0 2 -2 0 0 0 0 2 -2 -2 2 0 0 -2i 0 2i 2i -2i 0 0 0 complex lifted from C4○D4 ρ16 2 -2 2 -2 0 0 2 -2 0 0 0 0 -2 2 2 -2 0 0 0 -2i 0 0 0 -2i 2i 2i complex lifted from C4○D4 ρ17 2 -2 2 -2 0 0 2 -2 0 0 0 0 2 -2 -2 2 0 0 2i 0 -2i -2i 2i 0 0 0 complex lifted from C4○D4 ρ18 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 2 -2 √-2 √-2 √-2 -√-2 -√-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ19 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 2 -2 -√-2 -√-2 -√-2 √-2 √-2 √-2 -√-2 √-2 complex lifted from SD16 ρ20 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 -2 2 -√-2 √-2 -√-2 √-2 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ21 2 -2 2 -2 0 0 -2 2 0 0 0 0 0 0 0 0 -2 2 √-2 -√-2 √-2 -√-2 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ22 2 -2 2 -2 0 0 2 -2 0 0 0 0 -2 2 2 -2 0 0 0 2i 0 0 0 2i -2i -2i complex lifted from C4○D4 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 2√2 -2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 -2√2 2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22

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

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

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

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

Matrix representation of C8.13SD16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 9 13 3 0 0 8 8 14 13 0 0 3 13 9 8 0 0 4 3 9 9
,
 12 5 0 0 0 0 12 12 0 0 0 0 0 0 4 3 9 9 0 0 14 4 8 9 0 0 14 0 13 14 0 0 0 14 3 13
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 3 3 0 0 0 0 3 14

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

C8.13SD16 in GAP, Magma, Sage, TeX

C_8._{13}{\rm SD}_{16}
% in TeX

G:=Group("C8.13SD16");
// GroupNames label

G:=SmallGroup(128,976);
// by ID

G=gap.SmallGroup(128,976);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,568,422,723,58,1123,360,3924,102,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=a^6*b^3>;
// generators/relations

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