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G = C87D8order 128 = 27

1st semidirect product of C8 and D8 acting via D8/D4=C2

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

Aliases: C87D8, D41D8, C42.218C23, (C8×D4)⋊5C2, C81C83C2, C84D43C2, C4⋊D85C2, C4.59(C2×D8), C4⋊C4.193D4, (C2×C8).339D4, C4.D85C2, (C2×D4).187D4, C4.84(C4○D8), C2.6(C87D4), C2.7(C4⋊D8), C4⋊C8.19C22, (C4×C8).49C22, C4.116(C8⋊C22), C2.6(D4.4D4), (C4×D4).275C22, C41D4.26C22, C22.179(C4⋊D4), (C2×C4).3(C4○D4), (C2×C4).1253(C2×D4), SmallGroup(128,399)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C87D8
C1C2C22C2×C4C42C4×D4C8×D4 — C87D8
C1C22C42 — C87D8
C1C22C42 — C87D8
C1C22C22C42 — C87D8

Generators and relations for C87D8
 G = < a,b,c | a8=b8=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 304 in 102 conjugacy classes, 36 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×2], C22, C22 [×10], C8 [×2], C8 [×4], C2×C4 [×3], C2×C4 [×3], D4 [×2], D4 [×11], C23 [×3], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×5], 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], C22×C8, C2×D8 [×4], C4.D8 [×2], C81C8, C8×D4, C4⋊D8 [×2], C84D4, C87D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×4], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8 [×2], C4○D8, C8⋊C22, C4⋊D8, C87D4, D4.4D4, C87D8

Character table of C87D8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111441616222244422224444448888
ρ111111111111111111111111111111    trivial
ρ2111111-111111111-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ31111111-11111111-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ4111111-1-111111111111111111-1-1-1-1    linear of order 2
ρ51111-1-1-111111-1-11-1-1-1-1111-1-11-11-11    linear of order 2
ρ61111-1-1111111-1-111111-1-1-111-1-1-1-1-1    linear of order 2
ρ71111-1-1-1-11111-1-111111-1-1-111-11111    linear of order 2
ρ81111-1-11-11111-1-11-1-1-1-1111-1-111-11-1    linear of order 2
ρ922220000-22-2200-22222000-2-200000    orthogonal lifted from D4
ρ102222-2-2002-22-222-200000000000000    orthogonal lifted from D4
ρ1122220000-22-2200-2-2-2-2-20002200000    orthogonal lifted from D4
ρ12222222002-22-2-2-2-200000000000000    orthogonal lifted from D4
ρ132-22-20000020-20002-22-200000022-2-2    orthogonal lifted from D8
ρ1422-2-2-220020-200002-2-222-2-2-2220000    orthogonal lifted from D8
ρ152-22-20000020-2000-22-220000002-2-22    orthogonal lifted from D8
ρ1622-2-22-20020-200002-2-22-222-22-20000    orthogonal lifted from D8
ρ172-22-20000020-20002-22-2000000-2-222    orthogonal lifted from D8
ρ182-22-20000020-2000-22-22000000-222-2    orthogonal lifted from D8
ρ1922-2-2-220020-20000-222-2-2222-2-20000    orthogonal lifted from D8
ρ2022-2-22-20020-20000-222-22-2-22-220000    orthogonal lifted from D8
ρ2122220000-2-2-2-20020000-2i-2i2i002i0000    complex lifted from C4○D4
ρ2222-2-20000-2020-2i2i02-2-22-2--2-22-2--20000    complex lifted from C4○D8
ρ2322-2-20000-20202i-2i0-222-2-2--2-2-22--20000    complex lifted from C4○D8
ρ2422-2-20000-2020-2i2i0-222-2--2-2--2-22-20000    complex lifted from C4○D8
ρ2522-2-20000-20202i-2i02-2-22--2-2--22-2-20000    complex lifted from C4○D8
ρ2622220000-2-2-2-200200002i2i-2i00-2i0000    complex lifted from C4○D4
ρ274-44-400000-40400000000000000000    orthogonal lifted from C8⋊C22
ρ284-4-44000000000002222-22-220000000000    orthogonal lifted from D4.4D4
ρ294-4-4400000000000-22-2222220000000000    orthogonal lifted from D4.4D4

Smallest permutation representation of C87D8
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 31 57 44 22 53 36 12)(2 30 58 43 23 52 37 11)(3 29 59 42 24 51 38 10)(4 28 60 41 17 50 39 9)(5 27 61 48 18 49 40 16)(6 26 62 47 19 56 33 15)(7 25 63 46 20 55 34 14)(8 32 64 45 21 54 35 13)
(2 8)(3 7)(4 6)(9 26)(10 25)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 19)(20 24)(21 23)(33 60)(34 59)(35 58)(36 57)(37 64)(38 63)(39 62)(40 61)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)

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

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

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

Matrix representation of C87D8 in GL4(𝔽17) generated by

1000
0100
001111
0030
,
31400
3300
00160
0011
,
1000
01600
0010
001616
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,11,3,0,0,11,0],[3,3,0,0,14,3,0,0,0,0,16,1,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;

C87D8 in GAP, Magma, Sage, TeX

C_8\rtimes_7D_8
% in TeX

G:=Group("C8:7D8");
// GroupNames label

G:=SmallGroup(128,399);
// by ID

G=gap.SmallGroup(128,399);
# by ID

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

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

Export

Character table of C87D8 in TeX

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