Copied to
clipboard

G = M4(2)⋊1C8order 128 = 27

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

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

Aliases: M4(2)⋊1C8, C42.15Q8, C42.419D4, C42.622C23, C81(C2×C8), C82C87C2, C81C811C2, C4.18(C4⋊C8), C8⋊C4.4C4, C4.27(C22×C8), (C22×C4).11Q8, C4⋊C8.266C22, C23.48(C4⋊C4), C22.12(C4⋊C8), (C4×C8).137C22, C42.121(C2×C4), (C22×C4).667D4, (C4×M4(2)).1C2, (C2×C4).18M4(2), C4.42(C2×M4(2)), C4.137(C8⋊C22), (C2×M4(2)).10C4, C4.131(C8.C22), (C2×C42).224C22, C2.1(M4(2).C4), C2.1(M4(2)⋊C4), C42.12C4.29C2, C2.7(C2×C4⋊C8), (C2×C4⋊C8).18C2, (C2×C4).19(C2×C8), (C2×C8).56(C2×C4), C22.48(C2×C4⋊C4), (C2×C4).149(C2×Q8), (C2×C4).116(C4⋊C4), (C2×C4).1458(C2×D4), (C22×C4).246(C2×C4), (C2×C4).504(C22×C4), SmallGroup(128,297)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2)⋊1C8
C1C2C22C2×C4C42C2×C42C4×M4(2) — M4(2)⋊1C8
C1C2C4 — M4(2)⋊1C8
C1C2×C4C2×C42 — M4(2)⋊1C8
C1C22C22C42 — M4(2)⋊1C8

Generators and relations for M4(2)⋊1C8
 G = < a,b,c | a8=b2=c8=1, bab=a5, cac-1=a-1, bc=cb >

Subgroups: 132 in 90 conjugacy classes, 62 normal (32 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×3], C23, C42 [×4], C2×C8 [×4], C2×C8 [×6], M4(2) [×4], M4(2) [×2], C22×C4 [×3], C4×C8 [×2], C4×C8, C8⋊C4 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C22×C8, C2×M4(2) [×2], C82C8 [×2], C81C8 [×2], C4×M4(2), C2×C4⋊C8, C42.12C4, M4(2)⋊1C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C8⋊C22, C8.C22, C2×C4⋊C8, M4(2)⋊C4, M4(2).C4, M4(2)⋊1C8

Smallest permutation representation of M4(2)⋊1C8
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 43)(2 48)(3 45)(4 42)(5 47)(6 44)(7 41)(8 46)(9 59)(10 64)(11 61)(12 58)(13 63)(14 60)(15 57)(16 62)(17 25)(18 30)(19 27)(20 32)(21 29)(22 26)(23 31)(24 28)(33 51)(34 56)(35 53)(36 50)(37 55)(38 52)(39 49)(40 54)
(1 30 35 62 43 18 53 16)(2 29 36 61 44 17 54 15)(3 28 37 60 45 24 55 14)(4 27 38 59 46 23 56 13)(5 26 39 58 47 22 49 12)(6 25 40 57 48 21 50 11)(7 32 33 64 41 20 51 10)(8 31 34 63 42 19 52 9)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8X
order12222244444···48···8
size11112211112···24···4

44 irreducible representations

dim11111111122222444
type+++++++-+-+-
imageC1C2C2C2C2C2C4C4C8D4Q8D4Q8M4(2)C8⋊C22C8.C22M4(2).C4
kernelM4(2)⋊1C8C82C8C81C8C4×M4(2)C2×C4⋊C8C42.12C4C8⋊C4C2×M4(2)M4(2)C42C42C22×C4C22×C4C2×C4C4C4C2
# reps122111441611114112

Matrix representation of M4(2)⋊1C8 in GL6(𝔽17)

0160000
100000
00160150
0000161
0011610
000010
,
100000
010000
001000
000100
00160160
00160016
,
020000
200000
0091400
0016800
00710716
000101610

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,16,0,0,0,15,16,1,1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,9,16,7,0,0,0,14,8,10,10,0,0,0,0,7,16,0,0,0,0,16,10] >;

M4(2)⋊1C8 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_1C_8
% in TeX

G:=Group("M4(2):1C8");
// GroupNames label

G:=SmallGroup(128,297);
// by ID

G=gap.SmallGroup(128,297);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,387,1123,136,172]);
// Polycyclic

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

׿
×
𝔽