Copied to
clipboard

G = C87M4(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

Aliases: C87M4(2), C42.51Q8, C42.320D4, C42.624C23, (C4×C8).19C4, C81C825C2, (C2×C4).72D8, C4.83(C2×D8), (C2×C4).34Q16, C4.55(C2×Q16), (C22×C8).34C4, C4.14(C2.D8), C4.6(C8.C4), (C22×C4).82Q8, C4⋊C8.215C22, C23.50(C4⋊C4), C42.311(C2×C4), (C4×C8).390C22, (C22×C4).575D4, C4.44(C2×M4(2)), C22.12(C2.D8), C4⋊M4(2).21C2, C2.7(C4⋊M4(2)), (C2×C42).1042C22, (C2×C4×C8).32C2, C2.4(C2×C2.D8), (C2×C4).75(C4⋊C4), (C2×C8).220(C2×C4), C2.7(C2×C8.C4), C22.81(C2×C4⋊C4), (C2×C4).151(C2×Q8), (C2×C4).1460(C2×D4), (C2×C4).506(C22×C4), (C22×C4).475(C2×C4), SmallGroup(128,299)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C87M4(2)
C1C2C22C2×C4C42C2×C42C2×C4×C8 — C87M4(2)
C1C2C2×C4 — C87M4(2)
C1C2×C4C2×C42 — C87M4(2)
C1C22C22C42 — C87M4(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 2A2B2C2D2E4A4B4C4D4E···4N8A···8P8Q···8X
order12222244444···48···88···8
size11112211112···22···28···8

44 irreducible representations

dim11111122222222
type+++++-+-+-
imageC1C2C2C2C4C4D4Q8D4Q8M4(2)D8Q16C8.C4
kernelC87M4(2)C81C8C2×C4×C8C4⋊M4(2)C4×C8C22×C8C42C42C22×C4C22×C4C8C2×C4C2×C4C4
# reps14124411118448

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

8000
01500
0090
00152
,
0100
13000
001415
0043
,
1000
01600
0010
0001
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

׿
×
𝔽