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## G = C8⋊9SD16order 128 = 27

### 3rd semidirect product of C8 and SD16 acting via SD16/C8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8⋊9SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8⋊4Q8 — C8⋊9SD16
 Lower central C1 — C2 — C2×C4 — C8⋊9SD16
 Upper central C1 — C2×C4 — C4×C8 — C8⋊9SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊9SD16

Generators and relations for C89SD16
G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a5, cbc=b3 >

Subgroups: 152 in 83 conjugacy classes, 44 normal (40 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×4], C22, C22 [×3], C8 [×4], C8 [×6], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×2], C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×3], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C2×M4(2), C2×SD16, C82, D4⋊C8, Q8⋊C8, C82C8, C86D4, C4×SD16, C84Q8, C89SD16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×SD16, C4○D8, C86D4, C4×SD16, C8○D8, C89SD16

Smallest permutation representation of C89SD16
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 21 15 59 31 56 34 48)(2 22 16 60 32 49 35 41)(3 23 9 61 25 50 36 42)(4 24 10 62 26 51 37 43)(5 17 11 63 27 52 38 44)(6 18 12 64 28 53 39 45)(7 19 13 57 29 54 40 46)(8 20 14 58 30 55 33 47)
(2 6)(4 8)(9 36)(10 33)(11 38)(12 35)(13 40)(14 37)(15 34)(16 39)(17 63)(18 60)(19 57)(20 62)(21 59)(22 64)(23 61)(24 58)(26 30)(28 32)(41 53)(42 50)(43 55)(44 52)(45 49)(46 54)(47 51)(48 56)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A ··· 8X 8Y 8Z 8AA 8AB order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 size 1 1 1 1 8 1 1 1 1 2 2 2 2 8 8 8 2 ··· 2 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 M4(2) SD16 C4○D4 C8○D4 C4○D8 C8○D8 kernel C8⋊9SD16 C82 D4⋊C8 Q8⋊C8 C8⋊2C8 C8⋊6D4 C4×SD16 C8⋊4Q8 D4⋊C4 Q8⋊C4 C4.Q8 C2×SD16 C2×C8 C8 C8 C2×C4 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 2 4 4 8

Matrix representation of C89SD16 in GL4(𝔽17) generated by

 2 4 0 0 15 15 0 0 0 0 2 2 0 0 13 15
,
 10 10 0 0 12 0 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 16 16 0 0 0 0 1 0 0 0 15 16
G:=sub<GL(4,GF(17))| [2,15,0,0,4,15,0,0,0,0,2,13,0,0,2,15],[10,12,0,0,10,0,0,0,0,0,16,0,0,0,0,16],[1,16,0,0,0,16,0,0,0,0,1,15,0,0,0,16] >;

C89SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_9{\rm SD}_{16}
% in TeX

G:=Group("C8:9SD16");
// GroupNames label

G:=SmallGroup(128,322);
// by ID

G=gap.SmallGroup(128,322);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,723,100,1123,570,136,172]);
// 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^3>;
// generators/relations

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