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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

Aliases: C8.8SD16, C42.246C23, C4⋊C4.68D4, C82C818C2, (C2×C8).98D4, C82Q824C2, (C2×D4).62D4, C86D4.4C2, C4.62(C2×SD16), C4⋊Q8.67C22, C4⋊C8.188C22, (C4×C8).149C22, D42Q8.11C2, C4.6Q1618C2, C2.8(C8.D4), (C4×D4).48C22, C4.44(C8.C22), C2.11(D4.D4), C2.14(D4.4D4), C22.207(C4⋊D4), (C2×C4).31(C4○D4), (C2×C4).1281(C2×D4), SmallGroup(128,427)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8.8SD16
C1C2C22C2×C4C42C4×D4C86D4 — C8.8SD16
C1C22C42 — C8.8SD16
C1C22C42 — C8.8SD16
C1C22C22C42 — 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 [×3], C2, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×3], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×2], C23, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×3], M4(2) [×2], C22×C4, C2×D4, C2×Q8 [×2], C4×C8, C22⋊C8, D4⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C4×D4, C4⋊Q8 [×2], C2×M4(2), C4.6Q16 [×2], C82C8, C86D4, D42Q8 [×2], C82Q8, C8.8SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, SD16 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×SD16, C8.C22 [×3], D4.D4, C8.D4, D4.4D4, C8.8SD16

Character table of C8.8SD16

 class 12A2B2C2D4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 1111822224816164444888888
ρ111111111111111111111111    trivial
ρ211111111111-11-1-1-1-11-11-1-1-1    linear of order 2
ρ31111-111111-1111111-1-1-1-1-1-1    linear of order 2
ρ41111-111111-1-11-1-1-1-1-11-1111    linear of order 2
ρ51111-111111-1-1-11111111-1-11    linear of order 2
ρ61111-111111-11-1-1-1-1-11-1111-1    linear of order 2
ρ711111111111-1-11111-1-1-111-1    linear of order 2
ρ8111111111111-1-1-1-1-1-11-1-1-11    linear of order 2
ρ9222202-22-2-20002-2-22000000    orthogonal lifted from D4
ρ102222-2-22-22-22000000000000    orthogonal lifted from D4
ρ11222202-22-2-2000-222-2000000    orthogonal lifted from D4
ρ1222222-22-22-2-2000000000000    orthogonal lifted from D4
ρ1322220-2-2-2-2200000000002i-2i0    complex lifted from C4○D4
ρ1422220-2-2-2-220000000000-2i2i0    complex lifted from C4○D4
ρ152-2-220-202000000-220--2--2-200-2    complex lifted from SD16
ρ162-2-220-202000000-220-2-2--200--2    complex lifted from SD16
ρ172-2-220-2020000002-20--2-2-200--2    complex lifted from SD16
ρ182-2-220-2020000002-20-2--2--200-2    complex lifted from SD16
ρ1944-4-4000000000-220022000000    orthogonal lifted from D4.4D4
ρ2044-4-40000000002200-22000000    orthogonal lifted from D4.4D4
ρ214-44-400-40400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-4-44040-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-44-40040-400000000000000    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 43 64 55 22 38 16)(2 30 44 59 56 17 39 11)(3 25 45 62 49 20 40 14)(4 28 46 57 50 23 33 9)(5 31 47 60 51 18 34 12)(6 26 48 63 52 21 35 15)(7 29 41 58 53 24 36 10)(8 32 42 61 54 19 37 13)
(2 6)(4 8)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)(25 58)(26 63)(27 60)(28 57)(29 62)(30 59)(31 64)(32 61)(33 42)(34 47)(35 44)(36 41)(37 46)(38 43)(39 48)(40 45)(50 54)(52 56)

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

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

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

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

001600000
000160000
10000000
01000000
00003320
000014302
00000161414
000010314
,
125000000
1212000000
005120000
00550000
0000711415
0000110153
0000101167
00001771
,
10000000
016000000
00100000
000160000
00001000
00000100
00001414160
0000314016

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

Export

Character table of C8.8SD16 in TeX

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