Copied to
clipboard

G = M5(2)⋊7C4order 128 = 27

7th semidirect product of M5(2) and C4 acting via C4/C2=C2

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

Aliases: M5(2)⋊7C4, M4(2).3C8, C23.19M4(2), C4⋊C4.6C8, C4.8(C4×C8), (C2×C16)⋊10C4, C8.41(C4⋊C4), C4.17(C4⋊C8), (C2×C8).57Q8, (C2×C8).376D4, (C2×C4).56C42, C2.3(D4.C8), (C22×C16).2C2, C22.11(C4⋊C8), C4.32(C22⋊C8), C8.51(C22⋊C4), (C2×C4).40M4(2), C42⋊C2.14C4, C22.7(C8⋊C4), (C2×M5(2)).10C2, (C2×M4(2)).20C4, C82M4(2).13C2, C22.39(C22⋊C8), (C22×C8).572C22, C4.29(C2.C42), C2.17(C22.7C42), (C2×C4).47(C2×C8), (C2×C8).239(C2×C4), (C2×C4).107(C4⋊C4), (C22×C4).391(C2×C4), (C2×C4).386(C22⋊C4), SmallGroup(128,111)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M5(2)⋊7C4
C1C2C4C2×C4C2×C8C22×C8C82M4(2) — M5(2)⋊7C4
C1C2C4 — M5(2)⋊7C4
C1C2×C8C22×C8 — M5(2)⋊7C4
C1C2C2C2C2C4C4C22×C8 — M5(2)⋊7C4

Generators and relations for M5(2)⋊7C4
 G = < a,b,c | a16=b2=c4=1, bab=a9, cac-1=a13b, cbc-1=a8b >

Subgroups: 96 in 68 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×2], C23, C16 [×4], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×6], C2×C8, M4(2) [×2], M4(2), C22×C4, C4×C8, C8⋊C4, C2×C16 [×2], C2×C16 [×3], M5(2) [×2], M5(2), C42⋊C2, C22×C8, C2×M4(2), C82M4(2), C22×C16, C2×M5(2), M5(2)⋊7C4
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C22.7C42, D4.C8 [×2], M5(2)⋊7C4

Smallest permutation representation of M5(2)⋊7C4
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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)
(1 46 49 18)(2 43 50 31)(3 48 51 20)(4 45 52 17)(5 34 53 22)(6 47 54 19)(7 36 55 24)(8 33 56 21)(9 38 57 26)(10 35 58 23)(11 40 59 28)(12 37 60 25)(13 42 61 30)(14 39 62 27)(15 44 63 32)(16 41 64 29)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L8M8N8O8P16A···16P16Q···16X
order12222244444444448···88888888816···1616···16
size11112211112244441···1222244442···24···4

56 irreducible representations

dim111111111122222
type+++++-
imageC1C2C2C2C4C4C4C4C8C8D4Q8M4(2)M4(2)D4.C8
kernelM5(2)⋊7C4C82M4(2)C22×C16C2×M5(2)C2×C16M5(2)C42⋊C2C2×M4(2)C4⋊C4M4(2)C2×C8C2×C8C2×C4C23C2
# reps1111442288312216

Matrix representation of M5(2)⋊7C4 in GL3(𝔽17) generated by

900
0011
070
,
1600
010
0016
,
400
001
0160
G:=sub<GL(3,GF(17))| [9,0,0,0,0,7,0,11,0],[16,0,0,0,1,0,0,0,16],[4,0,0,0,0,16,0,1,0] >;

M5(2)⋊7C4 in GAP, Magma, Sage, TeX

M_5(2)\rtimes_7C_4
% in TeX

G:=Group("M5(2):7C4");
// GroupNames label

G:=SmallGroup(128,111);
// by ID

G=gap.SmallGroup(128,111);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,723,352,1018,136,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^9,c*a*c^-1=a^13*b,c*b*c^-1=a^8*b>;
// generators/relations

׿
×
𝔽