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## G = C8⋊12SD16order 128 = 27

### 3rd semidirect product of C8 and SD16 acting via SD16/D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8⋊12SD16
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C8×D4 — C8⋊12SD16
 Lower central C1 — C2 — C2×C4 — C8⋊12SD16
 Upper central C1 — C2×C4 — C4×C8 — C8⋊12SD16
 Jennings C1 — C22 — C22 — C42 — C8⋊12SD16

Generators and relations for C812SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a5, ac=ca, cbc=b3 >

Subgroups: 152 in 84 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C4×D4, C4×Q8, C22×C8, C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C82C8, C8×D4, C4×SD16, C84Q8, C812SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), SD16, C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C2×SD16, C4○D8, C89D4, C4×SD16, C8.26D4, C812SD16

Smallest permutation representation of C812SD16
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 46 36 62 31 20 12 49)(2 43 37 59 32 17 13 54)(3 48 38 64 25 22 14 51)(4 45 39 61 26 19 15 56)(5 42 40 58 27 24 16 53)(6 47 33 63 28 21 9 50)(7 44 34 60 29 18 10 55)(8 41 35 57 30 23 11 52)
(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E ··· 8R 8S 8T order 1 2 2 2 2 2 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 4 4 1 1 1 1 2 2 2 2 4 4 8 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⋊12SD16 C8⋊C8 D4⋊C8 Q8⋊C8 C8⋊2C8 C8×D4 C4×SD16 C8⋊4Q8 D4⋊C4 Q8⋊C4 C4.Q8 C2×SD16 C2×C8 C8 C2×C4 D4 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 C812SD16 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 8 0 0 0 0 9
,
 12 5 0 0 12 12 0 0 0 0 0 8 0 0 2 0
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,8,0,0,0,0,9],[12,12,0,0,5,12,0,0,0,0,0,2,0,0,8,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C812SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_{12}{\rm SD}_{16}
% in TeX

G:=Group("C8:12SD16");
// GroupNames label

G:=SmallGroup(128,314);
// by ID

G=gap.SmallGroup(128,314);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,268,1123,570,136,172]);
// Polycyclic

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

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