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G = D85C8order 128 = 27

5th semidirect product of D8 and C8 acting via C8/C4=C2

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

Aliases: D85C8, C42.637C23, C83(C2×C8), D4⋊C85C2, (C8×D4)⋊2C2, D42(C2×C8), C8⋊C83C2, C2.9(C8×D4), C82C811C2, (C4×D8).15C2, (C2×D8).11C4, (C2×C8).306D4, C2.D8.19C4, C4.12(C8○D4), C4.12(C22×C8), (C4×C8).13C22, C22.82(C4×D4), D4⋊C4.10C4, C2.1(D8⋊C4), C2.3(C8.26D4), C4⋊C8.275C22, C4.142(C8⋊C22), (C4×D4).271C22, C4⋊C4.131(C2×C4), (C2×C8).125(C2×C4), (C2×D4).151(C2×C4), (C2×C4).1473(C2×D4), (C2×C4).498(C4○D4), (C2×C4).329(C22×C4), SmallGroup(128,312)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D85C8
C1C2C22C2×C4C42C4×C8C8×D4 — D85C8
C1C2C4 — D85C8
C1C2×C4C4×C8 — D85C8
C1C22C22C42 — D85C8

Generators and relations for D85C8
 G = < a,b,c | a8=b2=c8=1, bab=a-1, cac-1=a5, cbc-1=a4b >

Subgroups: 184 in 96 conjugacy classes, 50 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×6], D4 [×4], D4 [×2], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×7], D8 [×4], C22×C4 [×2], C2×D4 [×2], C4×C8 [×3], C22⋊C8 [×2], D4⋊C4 [×2], C4⋊C8 [×2], C2.D8, C4×D4 [×2], C22×C8 [×2], C2×D8, C8⋊C8, D4⋊C8 [×2], C82C8, C8×D4 [×2], C4×D8, D85C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8⋊C22 [×2], C8×D4, D8⋊C4, C8.26D4, D85C8

Smallest permutation representation of D85C8
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 28)(2 27)(3 26)(4 25)(5 32)(6 31)(7 30)(8 29)(9 33)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 51)(18 50)(19 49)(20 56)(21 55)(22 54)(23 53)(24 52)(41 62)(42 61)(43 60)(44 59)(45 58)(46 57)(47 64)(48 63)
(1 38 44 55 28 16 59 17)(2 35 45 52 29 13 60 22)(3 40 46 49 30 10 61 19)(4 37 47 54 31 15 62 24)(5 34 48 51 32 12 63 21)(6 39 41 56 25 9 64 18)(7 36 42 53 26 14 57 23)(8 33 43 50 27 11 58 20)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L8A···8H8I···8X
order122222224444444444448···88···8
size111144441111222244442···24···4

44 irreducible representations

dim111111111122244
type++++++++
imageC1C2C2C2C2C2C4C4C4C8D4C4○D4C8○D4C8⋊C22C8.26D4
kernelD85C8C8⋊C8D4⋊C8C82C8C8×D4C4×D8D4⋊C4C2.D8C2×D8D8C2×C8C2×C4C4C4C2
# reps1121214221622422

Matrix representation of D85C8 in GL6(𝔽17)

140000
8160000
00314116
0031406
003300
003030
,
16130000
010000
001000
0001600
0000160
00016161
,
800000
080000
000010
00161115
001000
0000016

G:=sub<GL(6,GF(17))| [1,8,0,0,0,0,4,16,0,0,0,0,0,0,3,3,3,3,0,0,14,14,3,0,0,0,11,0,0,3,0,0,6,6,0,0],[16,0,0,0,0,0,13,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,15,0,16] >;

D85C8 in GAP, Magma, Sage, TeX

D_8\rtimes_5C_8
% in TeX

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

G:=SmallGroup(128,312);
// by ID

G=gap.SmallGroup(128,312);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,268,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b>;
// generators/relations

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