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G = C82D8order 128 = 27

2nd semidirect product of C8 and D8 acting via D8/C4=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C82D8, C42.238C23, C4⋊D86C2, C86D42C2, C4⋊C4.60D4, C82C814C2, (C2×C8).90D4, C4.62(C2×D8), C84D414C2, (C2×D4).58D4, C4.D810C2, C2.6(C82D4), C4⋊C8.29C22, C2.10(C4⋊D8), C4.69(C8⋊C22), (C4×C8).141C22, (C4×D4).44C22, C41D4.34C22, C2.12(D4.4D4), C22.199(C4⋊D4), (C2×C4).23(C4○D4), (C2×C4).1273(C2×D4), SmallGroup(128,419)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C82D8
C1C2C22C2×C4C42C4×D4C86D4 — C82D8
C1C22C42 — C82D8
C1C22C42 — C82D8
C1C22C22C42 — C82D8

Generators and relations for C82D8
 G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a-1, cbc=b-1 >

Subgroups: 304 in 99 conjugacy classes, 34 normal (22 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×9], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×3], D4 [×12], C23 [×3], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×3], M4(2) [×2], D8 [×8], C22×C4, C2×D4, C2×D4 [×6], C4×C8, C22⋊C8, D4⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4×D4, C41D4 [×2], C2×M4(2), C2×D8 [×4], C4.D8 [×2], C82C8, C86D4, C4⋊D8 [×2], C84D4, C82D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C8⋊C22 [×3], C4⋊D8, C82D4, D4.4D4, C82D8

Character table of C82D8

 class 12A2B2C2D2E2F4A4B4C4D4E4F8A8B8C8D8E8F8G8H8I8J
 size 1111816162222484444888888
ρ111111111111111111111111    trivial
ρ21111-1-1-111111-111111-111-11    linear of order 2
ρ311111-11111111-1-1-1-1-1-111-1-1    linear of order 2
ρ41111-11-111111-1-1-1-1-1-11111-1    linear of order 2
ρ51111-1-1111111-1-1-1-1-111-1-111    linear of order 2
ρ6111111-1111111-1-1-1-11-1-1-1-11    linear of order 2
ρ71111-11111111-11111-1-1-1-1-1-1    linear of order 2
ρ811111-1-11111111111-11-1-11-1    linear of order 2
ρ92222-200-22-22-220000000000    orthogonal lifted from D4
ρ1022220002-22-2-202-2-22000000    orthogonal lifted from D4
ρ112222200-22-22-2-20000000000    orthogonal lifted from D4
ρ1222220002-22-2-20-222-2000000    orthogonal lifted from D4
ρ132-2-22000-20200002-20202-20-2    orthogonal lifted from D8
ρ142-2-22000-2020000-220-202-202    orthogonal lifted from D8
ρ152-2-22000-2020000-22020-220-2    orthogonal lifted from D8
ρ162-2-22000-20200002-20-20-2202    orthogonal lifted from D8
ρ172222000-2-2-2-22000000-2i002i0    complex lifted from C4○D4
ρ182222000-2-2-2-220000002i00-2i0    complex lifted from C4○D4
ρ194-4-4400040-40000000000000    orthogonal lifted from C8⋊C22
ρ204-44-40000-404000000000000    orthogonal lifted from C8⋊C22
ρ214-44-4000040-4000000000000    orthogonal lifted from C8⋊C22
ρ2244-4-40000000002200-22000000    orthogonal lifted from D4.4D4
ρ2344-4-4000000000-220022000000    orthogonal lifted from D4.4D4

Smallest permutation representation of C82D8
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 23 37 12 55 30 57 45)(2 18 38 15 56 25 58 48)(3 21 39 10 49 28 59 43)(4 24 40 13 50 31 60 46)(5 19 33 16 51 26 61 41)(6 22 34 11 52 29 62 44)(7 17 35 14 53 32 63 47)(8 20 36 9 54 27 64 42)
(2 8)(3 7)(4 6)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 43)(18 42)(19 41)(20 48)(21 47)(22 46)(23 45)(24 44)(33 61)(34 60)(35 59)(36 58)(37 57)(38 64)(39 63)(40 62)(49 53)(50 52)(54 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), (1,23,37,12,55,30,57,45)(2,18,38,15,56,25,58,48)(3,21,39,10,49,28,59,43)(4,24,40,13,50,31,60,46)(5,19,33,16,51,26,61,41)(6,22,34,11,52,29,62,44)(7,17,35,14,53,32,63,47)(8,20,36,9,54,27,64,42), (2,8)(3,7)(4,6)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,43)(18,42)(19,41)(20,48)(21,47)(22,46)(23,45)(24,44)(33,61)(34,60)(35,59)(36,58)(37,57)(38,64)(39,63)(40,62)(49,53)(50,52)(54,56)>;

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

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

Matrix representation of C82D8 in GL6(𝔽17)

1600000
0160000
00161143
0016161414
00314116
003311
,
060000
1460000
000010
0000016
001000
0001600
,
100000
1160000
001000
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,16,3,3,0,0,1,16,14,3,0,0,14,14,1,1,0,0,3,14,16,1],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C82D8 in GAP, Magma, Sage, TeX

C_8\rtimes_2D_8
% in TeX

G:=Group("C8:2D8");
// GroupNames label

G:=SmallGroup(128,419);
// by ID

G=gap.SmallGroup(128,419);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,387,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C82D8 in TeX

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