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G = C85D8order 128 = 27

2nd semidirect product of C8 and D8 acting via D8/C8=C2

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

Aliases: C85D8, C827C2, C87SD16, C42.653C23, C4.1(C2×D8), C82Q86C2, C85D417C2, C4.1(C4○D8), C84D4.5C2, C4.4D87C2, (C2×C8).220D4, C4.2(C2×SD16), C2.6(C85D4), C2.4(C84D4), C4⋊Q8.78C22, (C4×C8).427C22, C2.6(C8.12D4), C41D4.41C22, C22.54(C41D4), (C2×C4).710(C2×D4), SmallGroup(128,438)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C85D8
C1C2C22C2×C4C42C4×C8C82 — C85D8
C1C22C42 — C85D8
C1C22C42 — C85D8
C1C22C22C42 — C85D8

Generators and relations for C85D8
 G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a3, cbc=b-1 >

Subgroups: 320 in 112 conjugacy classes, 44 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×2], C22, C22 [×6], C8 [×8], C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×12], Q8 [×4], C23 [×2], C42, C4⋊C4 [×4], C2×C8 [×6], D8 [×4], SD16 [×8], C2×D4 [×6], C2×Q8 [×2], C4×C8 [×3], D4⋊C4 [×4], C2.D8 [×2], C41D4 [×2], C4⋊Q8 [×2], C2×D8 [×2], C2×SD16 [×4], C82, C4.4D8 [×2], C85D4 [×2], C84D4, C82Q8, C85D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], SD16 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C2×SD16 [×2], C4○D8 [×2], C85D4, C84D4, C8.12D4, C85D8

Smallest permutation representation of C85D8
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 63 23 29 47 38 16 54)(2 64 24 30 48 39 9 55)(3 57 17 31 41 40 10 56)(4 58 18 32 42 33 11 49)(5 59 19 25 43 34 12 50)(6 60 20 26 44 35 13 51)(7 61 21 27 45 36 14 52)(8 62 22 28 46 37 15 53)
(1 54)(2 49)(3 52)(4 55)(5 50)(6 53)(7 56)(8 51)(9 58)(10 61)(11 64)(12 59)(13 62)(14 57)(15 60)(16 63)(17 36)(18 39)(19 34)(20 37)(21 40)(22 35)(23 38)(24 33)(25 43)(26 46)(27 41)(28 44)(29 47)(30 42)(31 45)(32 48)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H8A···8X
order1222224···4448···8
size111116162···216162···2

38 irreducible representations

dim1111112222
type++++++++
imageC1C2C2C2C2C2D4D8SD16C4○D8
kernelC85D8C82C4.4D8C85D4C84D4C82Q8C2×C8C8C8C4
# reps1122116888

Matrix representation of C85D8 in GL4(𝔽17) generated by

1000
0100
0055
00125
,
31400
3300
0033
00143
,
14300
3300
0033
00314
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,5,12,0,0,5,5],[3,3,0,0,14,3,0,0,0,0,3,14,0,0,3,3],[14,3,0,0,3,3,0,0,0,0,3,3,0,0,3,14] >;

C85D8 in GAP, Magma, Sage, TeX

C_8\rtimes_5D_8
% in TeX

G:=Group("C8:5D8");
// GroupNames label

G:=SmallGroup(128,438);
// by ID

G=gap.SmallGroup(128,438);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,100,1123,136,2804,172]);
// Polycyclic

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

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