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## G = C8⋊7M4(2)  order 128 = 27

### 1st semidirect product of C8 and M4(2) acting via M4(2)/C2×C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8⋊7M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4×C8 — C8⋊7M4(2)
 Lower central C1 — C2 — C2×C4 — C8⋊7M4(2)
 Upper central C1 — C2×C4 — C2×C42 — C8⋊7M4(2)
 Jennings C1 — C22 — C22 — C42 — C8⋊7M4(2)

Generators and relations for C87M4(2)
G = < a,b,c | a8=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >

Subgroups: 140 in 92 conjugacy classes, 60 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×4], C2×C8 [×8], M4(2) [×4], C22×C4 [×3], C4×C8 [×2], C4×C8 [×2], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C81C8 [×4], C2×C4×C8, C4⋊M4(2) [×2], C87M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, C2.D8 [×4], C8.C4 [×2], C2×C4⋊C4, C2×M4(2) [×2], C2×D8, C2×Q16, C4⋊M4(2), C2×C2.D8, C2×C8.C4, C87M4(2)

Smallest permutation representation of C87M4(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 22 43 26 39 60 52 13)(2 21 44 25 40 59 53 12)(3 20 45 32 33 58 54 11)(4 19 46 31 34 57 55 10)(5 18 47 30 35 64 56 9)(6 17 48 29 36 63 49 16)(7 24 41 28 37 62 50 15)(8 23 42 27 38 61 51 14)
(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 8A ··· 8P 8Q ··· 8X order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 2 ··· 2 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + - + - image C1 C2 C2 C2 C4 C4 D4 Q8 D4 Q8 M4(2) D8 Q16 C8.C4 kernel C8⋊7M4(2) C8⋊1C8 C2×C4×C8 C4⋊M4(2) C4×C8 C22×C8 C42 C42 C22×C4 C22×C4 C8 C2×C4 C2×C4 C4 # reps 1 4 1 2 4 4 1 1 1 1 8 4 4 8

Matrix representation of C87M4(2) in GL4(𝔽17) generated by

 8 0 0 0 0 15 0 0 0 0 9 0 0 0 15 2
,
 0 1 0 0 13 0 0 0 0 0 14 15 0 0 4 3
,
 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(17))| [8,0,0,0,0,15,0,0,0,0,9,15,0,0,0,2],[0,13,0,0,1,0,0,0,0,0,14,4,0,0,15,3],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1] >;

C87M4(2) in GAP, Magma, Sage, TeX

C_8\rtimes_7M_4(2)
% in TeX

G:=Group("C8:7M4(2)");
// GroupNames label

G:=SmallGroup(128,299);
// by ID

G=gap.SmallGroup(128,299);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,288,1430,1123,136,172]);
// Polycyclic

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

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