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## G = C8⋊15SD16order 128 = 27

### 3rd semidirect product of C8 and SD16 acting via SD16/Q8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8⋊15SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8×Q8 — C8⋊15SD16
 Lower central C1 — C2 — C2×C4 — C8⋊15SD16
 Upper central C1 — C2×C4 — C4×C8 — C8⋊15SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊15SD16

Generators and relations for C815SD16
G = < a,b,c | a8=b8=c2=1, bab-1=cac=a5, cbc=b3 >

Subgroups: 152 in 83 conjugacy classes, 44 normal (40 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×5], C22, C22 [×3], C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×2], Q8, 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], C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C2×M4(2), C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C82C8, C86D4, C4×SD16, C8×Q8, C815SD16
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, C89D4, C4×SD16, C8.26D4, C815SD16

Smallest permutation representation of C815SD16
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 23 37 63 15 44 25 54)(2 20 38 60 16 41 26 51)(3 17 39 57 9 46 27 56)(4 22 40 62 10 43 28 53)(5 19 33 59 11 48 29 50)(6 24 34 64 12 45 30 55)(7 21 35 61 13 42 31 52)(8 18 36 58 14 47 32 49)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 56)(18 53)(19 50)(20 55)(21 52)(22 49)(23 54)(24 51)(26 30)(28 32)(34 38)(36 40)(41 64)(42 61)(43 58)(44 63)(45 60)(46 57)(47 62)(48 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), (1,23,37,63,15,44,25,54)(2,20,38,60,16,41,26,51)(3,17,39,57,9,46,27,56)(4,22,40,62,10,43,28,53)(5,19,33,59,11,48,29,50)(6,24,34,64,12,45,30,55)(7,21,35,61,13,42,31,52)(8,18,36,58,14,47,32,49), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,56)(18,53)(19,50)(20,55)(21,52)(22,49)(23,54)(24,51)(26,30)(28,32)(34,38)(36,40)(41,64)(42,61)(43,58)(44,63)(45,60)(46,57)(47,62)(48,59)>;

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 8A 8B 8C 8D 8E ··· 8R 8S 8T order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 size 1 1 1 1 8 1 1 1 1 2 2 2 2 4 4 4 4 8 2 2 2 2 4 ··· 4 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 SD16 C4○D4 M4(2) C8○D4 C4○D8 C8.26D4 kernel C8⋊15SD16 C8⋊C8 D4⋊C8 Q8⋊C8 C8⋊2C8 C8⋊6D4 C4×SD16 C8×Q8 D4⋊C4 Q8⋊C4 C4.Q8 C2×SD16 C2×C8 C8 C2×C4 Q8 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 4 2 4 4 4 2

Matrix representation of C815SD16 in GL4(𝔽17) generated by

 4 0 0 0 0 4 0 0 0 0 14 15 0 0 15 3
,
 12 12 0 0 5 12 0 0 0 0 16 0 0 0 3 1
,
 16 0 0 0 0 1 0 0 0 0 1 0 0 0 14 16
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,14,15,0,0,15,3],[12,5,0,0,12,12,0,0,0,0,16,3,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,1,14,0,0,0,16] >;

C815SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_{15}{\rm SD}_{16}
% in TeX

G:=Group("C8:15SD16");
// GroupNames label

G:=SmallGroup(128,315);
// by ID

G=gap.SmallGroup(128,315);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,100,1123,570,136,172]);
// Polycyclic

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

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