Copied to
clipboard

## G = C8.12SD16order 128 = 27

### 12nd 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 — C8.12SD16
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C16⋊5C4 — C8.12SD16
 Lower central C1 — C2 — C4 — C2×C8 — C8.12SD16
 Upper central C1 — C22 — C42 — C4×C8 — C8.12SD16
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.12SD16

Generators and relations for C8.12SD16
G = < a,b,c | a8=c2=1, b8=a4, bab-1=a5, cac=a3, cbc=a6b3 >

Subgroups: 200 in 71 conjugacy classes, 32 normal (20 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×4], C23, C16 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], D8 [×2], SD16 [×2], Q16 [×2], C2×D4, C2×Q8 [×2], C4×C8, C2.D8 [×2], C2.D8, C2×C16 [×2], C4.4D4, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C165C4, C2.D16 [×2], C2.Q32 [×2], C8.12D4, C82Q8, C8.12SD16
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, Q32⋊C2, C8.12SD16

Character table of C8.12SD16

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 16 2 2 4 4 16 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 2 2 -2 -2 0 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 2 2 2 2 0 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 -2 -2 -2 2 0 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 -2 -2 -2 2 0 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 -2 -2 2 -2 0 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 -2 -2 2 -2 0 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 2 -2 0 0 0 0 0 2 -2 2 -2 0 0 2i 0 -2i 0 2i -2i 0 0 complex lifted from C4○D4 ρ16 2 -2 2 -2 0 -2 2 0 0 0 0 0 0 0 0 0 2 -2 -√-2 √-2 -√-2 -√-2 √-2 √-2 -√-2 √-2 complex lifted from SD16 ρ17 2 -2 2 -2 0 2 -2 0 0 0 0 0 -2 2 -2 2 0 0 0 2i 0 2i 0 0 -2i -2i complex lifted from C4○D4 ρ18 2 -2 2 -2 0 2 -2 0 0 0 0 0 -2 2 -2 2 0 0 0 -2i 0 -2i 0 0 2i 2i complex lifted from C4○D4 ρ19 2 -2 2 -2 0 2 -2 0 0 0 0 0 2 -2 2 -2 0 0 -2i 0 2i 0 -2i 2i 0 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 0 -2 2 0 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 -2 2 0 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 -2 2 0 0 0 0 0 0 0 0 0 2 -2 √-2 -√-2 √-2 √-2 -√-2 -√-2 √-2 -√-2 complex lifted from SD16 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 symplectic lifted from Q32⋊C2, Schur index 2 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 symplectic lifted from Q32⋊C2, Schur index 2

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

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

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

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

Matrix representation of C8.12SD16 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 7 1 12 8 0 0 1 9 15 7 0 0 10 6 5 13 0 0 15 2 12 13
,
 12 5 0 0 0 0 12 12 0 0 0 0 0 0 16 6 9 8 0 0 10 3 8 0 0 0 12 14 7 7 0 0 15 9 0 8
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 16 16 0 0 0 0 9 15 14 14 0 0 6 14 14 3

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,1,10,15,0,0,1,9,6,2,0,0,12,15,5,12,0,0,8,7,13,13],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,16,10,12,15,0,0,6,3,14,9,0,0,9,8,7,0,0,0,8,0,7,8],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,16,9,6,0,0,0,16,15,14,0,0,0,0,14,14,0,0,0,0,14,3] >;

C8.12SD16 in GAP, Magma, Sage, TeX

C_8._{12}{\rm SD}_{16}
% in TeX

G:=Group("C8.12SD16");
// GroupNames label

G:=SmallGroup(128,975);
// by ID

G=gap.SmallGroup(128,975);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,120,422,723,58,1123,360,3924,102,4037,124]);
// 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^3,c*b*c=a^6*b^3>;
// generators/relations

Export

׿
×
𝔽