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## G = C8⋊9M4(2)  order 128 = 27

### 3rd semidirect product of C8 and M4(2) acting via M4(2)/C2×C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C8⋊9M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×C8 — C8⋊9M4(2)
 Lower central C1 — C22 — C8⋊9M4(2)
 Upper central C1 — C42 — C8⋊9M4(2)
 Jennings C1 — C22 — C22 — C42 — C8⋊9M4(2)

Generators and relations for C89M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a5, ac=ca, cbc=b5 >

Subgroups: 132 in 104 conjugacy classes, 76 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×10], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×12], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C4×C8 [×2], C4×C8 [×6], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C8⋊C8 [×4], C2×C4×C8, C4×M4(2), C42.12C4, C89M4(2)
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×8], C22×C4 [×3], C8⋊C4 [×4], C2×C42, C2×M4(2) [×4], C8○D4 [×2], C2×C8⋊C4, C4×M4(2), C82M4(2), C89M4(2)

Smallest permutation representation of C89M4(2)
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 19 36 54 13 43 25 59)(2 24 37 51 14 48 26 64)(3 21 38 56 15 45 27 61)(4 18 39 53 16 42 28 58)(5 23 40 50 9 47 29 63)(6 20 33 55 10 44 30 60)(7 17 34 52 11 41 31 57)(8 22 35 49 12 46 32 62)
(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 8A ··· 8P 8Q ··· 8AF order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C4 C4 M4(2) M4(2) C8○D4 kernel C8⋊9M4(2) C8⋊C8 C2×C4×C8 C4×M4(2) C42.12C4 C4×C8 C8⋊C4 C22⋊C8 C4⋊C8 C22×C8 C2×M4(2) C8 C2×C4 C4 # reps 1 4 1 1 1 4 4 4 4 4 4 8 8 8

Matrix representation of C89M4(2) in GL4(𝔽17) generated by

 4 0 0 0 0 4 0 0 0 0 0 1 0 0 4 0
,
 5 15 0 0 6 12 0 0 0 0 7 2 0 0 9 10
,
 1 0 0 0 5 16 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,4,0,0,1,0],[5,6,0,0,15,12,0,0,0,0,7,9,0,0,2,10],[1,5,0,0,0,16,0,0,0,0,1,0,0,0,0,1] >;

C89M4(2) in GAP, Magma, Sage, TeX

C_8\rtimes_9M_4(2)
% in TeX

G:=Group("C8:9M4(2)");
// GroupNames label

G:=SmallGroup(128,183);
// by ID

G=gap.SmallGroup(128,183);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,925,120,758,136,172]);
// 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^5>;
// generators/relations

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