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G = C8.13SD16  order 128 = 27

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

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

Aliases: C8.13SD16, C42.154D4, (C2×C4).46D8, C16⋊5C4⋊13C2, (C2×C8).136D4, C2.D16⋊19C2, C8.54(C4○D4), C8⋊4D4.15C2, C8.5Q8⋊13C2, C4.18(C2×SD16), (C2×C8).542C23, (C2×C16).59C22, (C4×C8).168C22, (C2×D8).15C22, C22.128(C2×D8), C2.21(C16⋊C22), C2.D8.27C22, C2.13(C4.4D8), C4.11(C4.4D4), (C2×C4).810(C2×D4), SmallGroup(128,976)

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, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C16, C42, C4⋊C4, C2×C8, D8, C2×D4, C4×C8, C4.Q8, C2.D8, C2×C16, C42.C2, C4⋊1D4, C2×D8, C2×D8, C16⋊5C4, C2.D16, C8⋊4D4, C8.5Q8, C8.13SD16
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, C4.4D4, C2×D8, C2×SD16, C4.4D8, C16⋊C22, C8.13SD16

Character table of C8.13SD16

 class 12A2B2C2D2E4A4B4C4D4E4F8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111116162244161622224444444444
ρ111111111111111111111111111    trivial
ρ211111-111-1-11-11111-1-1-11-1-1-1111    linear of order 2
ρ311111-111-1-1-111111-1-11-1111-1-1-1    linear of order 2
ρ41111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1111-1-11-11111-1-11-1111-1-1-1    linear of order 2
ρ61111-1-1111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-11111-1-111111111111111    linear of order 2
ρ81111-1111-1-1-111111-1-1-11-1-1-1111    linear of order 2
ρ922220022-2-200-2-2-2-22200000000    orthogonal lifted from D4
ρ10222200222200-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ11222200-2-2-2200000000√2√2-√2√2-√2-√2-√2√2    orthogonal lifted from D8
ρ12222200-2-22-200000000√2-√2-√2√2-√2√2√2-√2    orthogonal lifted from D8
ρ13222200-2-22-200000000-√2√2√2-√2√2-√2-√2√2    orthogonal lifted from D8
ρ14222200-2-2-2200000000-√2-√2√2-√2√2√2√2-√2    orthogonal lifted from D8
ρ152-22-2002-200002-2-2200-2i02i2i-2i000    complex lifted from C4○D4
ρ162-22-2002-20000-222-2000-2i000-2i2i2i    complex lifted from C4○D4
ρ172-22-2002-200002-2-22002i0-2i-2i2i000    complex lifted from C4○D4
ρ182-22-200-22000000002-2√-2√-2√-2-√-2-√-2-√-2√-2-√-2    complex lifted from SD16
ρ192-22-200-22000000002-2-√-2-√-2-√-2√-2√-2√-2-√-2√-2    complex lifted from SD16
ρ202-22-200-2200000000-22-√-2√-2-√-2√-2√-2-√-2√-2-√-2    complex lifted from SD16
ρ212-22-200-2200000000-22√-2-√-2√-2-√-2-√-2√-2-√-2√-2    complex lifted from SD16
ρ222-22-2002-20000-222-20002i0002i-2i-2i    complex lifted from C4○D4
ρ234-4-44000000002√22√2-2√2-2√20000000000    orthogonal lifted from C16⋊C22
ρ2444-4-4000000002√2-2√22√2-2√20000000000    orthogonal lifted from C16⋊C22
ρ254-4-4400000000-2√2-2√22√22√20000000000    orthogonal lifted from C16⋊C22
ρ2644-4-400000000-2√22√2-2√22√20000000000    orthogonal lifted from C16⋊C22

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

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

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

100000
010000
0089133
00881413
0031398
004399
,
1250000
12120000
004399
0014489
001401314
00014313
,
010000
100000
001000
0001600
000033
0000314

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
 

Export

Character table of C8.13SD16 in TeX

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