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G = SD16⋊C8order 128 = 27

The semidirect product of SD16 and C8 acting via C8/C4=C2

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

Aliases: SD16⋊C8, C42.635C23, C82(C2×C8), Q8⋊C85C2, Q82(C2×C8), (C8×Q8)⋊2C2, C8⋊C82C2, C2.7(C8×D4), D4⋊C8.1C2, C81C815C2, D4.2(C2×C8), (C8×D4).3C2, C4.Q8.5C4, (C2×C8).304D4, D4⋊C4.9C4, C4.10(C8○D4), (C4×C8).11C22, C4.10(C22×C8), Q8⋊C4.9C4, (C4×SD16).3C2, (C2×SD16).1C4, C22.80(C4×D4), C2.1(C8.26D4), C4⋊C8.273C22, C4.141(C8⋊C22), (C4×D4).270C22, (C4×Q8).257C22, C2.1(SD16⋊C4), C4.135(C8.C22), C4⋊C4.129(C2×C4), (C2×C8).123(C2×C4), (C2×D4).150(C2×C4), (C2×C4).1471(C2×D4), (C2×Q8).133(C2×C4), (C2×C4).496(C4○D4), (C2×C4).327(C22×C4), SmallGroup(128,310)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — SD16⋊C8
C1C2C22C2×C4C42C4×C8C8×D4 — SD16⋊C8
C1C2C4 — SD16⋊C8
C1C2×C4C4×C8 — SD16⋊C8
C1C22C22C42 — SD16⋊C8

Generators and relations for SD16⋊C8
 G = < a,b,c | a8=b2=c8=1, bab=a3, cac-1=a5, bc=cb >

Subgroups: 152 in 88 conjugacy classes, 50 normal (40 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×5], C22, C22 [×4], C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×2], Q8, C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×5], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C81C8, C8×D4, C4×SD16, C8×Q8, SD16⋊C8
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, C8.C22, C8×D4, SD16⋊C4, C8.26D4, SD16⋊C8

Smallest permutation representation of SD16⋊C8
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 64)(2 59)(3 62)(4 57)(5 60)(6 63)(7 58)(8 61)(9 18)(10 21)(11 24)(12 19)(13 22)(14 17)(15 20)(16 23)(25 52)(26 55)(27 50)(28 53)(29 56)(30 51)(31 54)(32 49)(33 42)(34 45)(35 48)(36 43)(37 46)(38 41)(39 44)(40 47)
(1 27 19 38 64 50 12 41)(2 32 20 35 57 55 13 46)(3 29 21 40 58 52 14 43)(4 26 22 37 59 49 15 48)(5 31 23 34 60 54 16 45)(6 28 24 39 61 51 9 42)(7 25 17 36 62 56 10 47)(8 30 18 33 63 53 11 44)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N8A···8H8I···8X
order122222444444444···48···88···8
size111144111122224···42···24···4

44 irreducible representations

dim1111111111111222444
type++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4C8D4C4○D4C8○D4C8⋊C22C8.C22C8.26D4
kernelSD16⋊C8C8⋊C8D4⋊C8Q8⋊C8C81C8C8×D4C4×SD16C8×Q8D4⋊C4Q8⋊C4C4.Q8C2×SD16SD16C2×C8C2×C4C4C4C4C2
# reps11111111222216224112

Matrix representation of SD16⋊C8 in GL6(𝔽17)

140000
8160000
0004016
00151391
0001013
0081624
,
1600000
910000
001000
00161600
000010
00001616
,
900000
090000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [1,8,0,0,0,0,4,16,0,0,0,0,0,0,0,15,0,8,0,0,4,13,1,16,0,0,0,9,0,2,0,0,16,1,13,4],[16,9,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

SD16⋊C8 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes C_8
% in TeX

G:=Group("SD16:C8");
// GroupNames label

G:=SmallGroup(128,310);
// by ID

G=gap.SmallGroup(128,310);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^2=c^8=1,b*a*b=a^3,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations

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