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

Aliases: C8.12SD16, C42.153D4, (C2×C4).45D8, C82Q826C2, C165C412C2, (C2×C8).135D4, C2.D16.7C2, C8.53(C4○D4), C4.17(C2×SD16), C2.Q3219C2, (C4×C8).167C22, (C2×C8).541C23, (C2×C16).58C22, C8.12D4.5C2, (C2×D8).14C22, C22.127(C2×D8), C2.20(C16⋊C22), C2.D8.26C22, C2.20(Q32⋊C2), C4.10(C4.4D4), C2.12(C4.4D8), (C2×Q16).15C22, (C2×C4).809(C2×D4), SmallGroup(128,975)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C8.12SD16
C1C2C4C8C2×C8C2×C16C165C4 — C8.12SD16
C1C2C4C2×C8 — C8.12SD16
C1C22C42C4×C8 — C8.12SD16
C1C2C2C2C2C4C4C2×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 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111116224416161622224444444444
ρ111111111111111111111111111    trivial
ρ21111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111-1-1-1-111111-1-11-11-111-1-1    linear of order 2
ρ41111111-1-1-11-11111-1-1-11-11-1-111    linear of order 2
ρ51111-111-1-11-111111-1-1-11-11-1-111    linear of order 2
ρ61111-111-1-111-11111-1-11-11-111-1-1    linear of order 2
ρ71111-11111-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-11111-1-1-111111111111111    linear of order 2
ρ92222022-2-2000-2-2-2-22200000000    orthogonal lifted from D4
ρ10222202222000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1122220-2-2-22000000000-2-2222-2-22    orthogonal lifted from D8
ρ1222220-2-2-2200000000022-2-2-222-2    orthogonal lifted from D8
ρ1322220-2-22-20000000002-2-22-22-22    orthogonal lifted from D8
ρ1422220-2-22-2000000000-222-22-22-2    orthogonal lifted from D8
ρ152-22-202-2000002-22-2002i0-2i02i-2i00    complex lifted from C4○D4
ρ162-22-20-220000000002-2--2-2--2--2-2-2--2-2    complex lifted from SD16
ρ172-22-202-200000-22-220002i02i00-2i-2i    complex lifted from C4○D4
ρ182-22-202-200000-22-22000-2i0-2i002i2i    complex lifted from C4○D4
ρ192-22-202-2000002-22-200-2i02i0-2i2i00    complex lifted from C4○D4
ρ202-22-20-22000000000-22--2--2--2-2-2-2-2--2    complex lifted from SD16
ρ212-22-20-22000000000-22-2-2-2--2--2--2--2-2    complex lifted from SD16
ρ222-22-20-220000000002-2-2--2-2-2--2--2-2--2    complex lifted from SD16
ρ234-4-4400000000-222222-220000000000    orthogonal lifted from C16⋊C22
ρ244-4-440000000022-22-22220000000000    orthogonal lifted from C16⋊C22
ρ2544-4-400000000-22-2222220000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2644-4-4000000002222-22-220000000000    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)

1600000
0160000
0071128
0019157
00106513
001521213
,
1250000
12120000
0016698
0010380
00121477
0015908
,
010000
100000
001000
00161600
009151414
00614143

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

Character table of C8.12SD16 in TeX

׿
×
𝔽