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G = C86D8order 128 = 27

3rd semidirect product of C8 and D8 acting via D8/C8=C2

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

Aliases: C86D8, C829C2, C84M4(2), C42.646C23, C2.5(C4×D8), D4⋊C835C2, (C2×D8).8C4, C81C830C2, (C4×D8).3C2, C4.87(C2×D8), C86D430C2, (C2×C8).284D4, C2.D8.13C4, D4⋊C4.1C4, C2.10(C8○D8), C4.13(C8○D4), C2.6(C86D4), C4.7(C2×M4(2)), C4.132(C4○D8), C4⋊C8.224C22, (C4×C8).392C22, (C4×D4).13C22, C22.137(C4×D4), C4⋊C4.59(C2×C4), (C2×C8).169(C2×C4), (C2×D4).58(C2×C4), (C2×C4).1482(C2×D4), (C2×C4).507(C4○D4), (C2×C4).338(C22×C4), SmallGroup(128,321)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C86D8
C1C2C22C2×C4C42C4×C8C86D4 — C86D8
C1C2C2×C4 — C86D8
C1C2×C4C4×C8 — C86D8
C1C22C22C42 — C86D8

Generators and relations for C86D8
 G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a5, cbc=b-1 >

Subgroups: 184 in 90 conjugacy classes, 44 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×3], C22, C22 [×6], C8 [×4], C8 [×6], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×3], M4(2) [×4], D8 [×2], C22×C4 [×2], C2×D4 [×2], C4×C8 [×3], C22⋊C8 [×2], D4⋊C4 [×2], C4⋊C8 [×2], C2.D8, C4×D4 [×2], C2×M4(2) [×2], C2×D8, C82, D4⋊C8 [×2], C81C8, C86D4 [×2], C4×D8, C86D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], D8 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×D8, C4○D8, C86D4, C4×D8, C8○D8, C86D8

Smallest permutation representation of C86D8
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 24 10 62 31 48 34 56)(2 17 11 63 32 41 35 49)(3 18 12 64 25 42 36 50)(4 19 13 57 26 43 37 51)(5 20 14 58 27 44 38 52)(6 21 15 59 28 45 39 53)(7 22 16 60 29 46 40 54)(8 23 9 61 30 47 33 55)
(1 56)(2 53)(3 50)(4 55)(5 52)(6 49)(7 54)(8 51)(9 43)(10 48)(11 45)(12 42)(13 47)(14 44)(15 41)(16 46)(17 39)(18 36)(19 33)(20 38)(21 35)(22 40)(23 37)(24 34)(25 64)(26 61)(27 58)(28 63)(29 60)(30 57)(31 62)(32 59)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8X8Y8Z8AA8AB
order12222244444444448···88888
size11118811112222882···28888

44 irreducible representations

dim1111111112222222
type++++++++
imageC1C2C2C2C2C2C4C4C4D4M4(2)D8C4○D4C8○D4C4○D8C8○D8
kernelC86D8C82D4⋊C8C81C8C86D4C4×D8D4⋊C4C2.D8C2×D8C2×C8C8C8C2×C4C4C4C2
# reps1121214222442448

Matrix representation of C86D8 in GL4(𝔽17) generated by

0800
9000
0082
0049
,
3300
14300
00164
0081
,
3300
31400
00164
0001
G:=sub<GL(4,GF(17))| [0,9,0,0,8,0,0,0,0,0,8,4,0,0,2,9],[3,14,0,0,3,3,0,0,0,0,16,8,0,0,4,1],[3,3,0,0,3,14,0,0,0,0,16,0,0,0,4,1] >;

C86D8 in GAP, Magma, Sage, TeX

C_8\rtimes_6D_8
% in TeX

G:=Group("C8:6D8");
// GroupNames label

G:=SmallGroup(128,321);
// by ID

G=gap.SmallGroup(128,321);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,723,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

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