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G = M5(2)⋊S3order 192 = 26·3

5th semidirect product of M5(2) and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.3D8, D24.2C4, C4.11D24, C24.80D4, M5(2)⋊5S3, C8.4(C4×S3), C24.1(C2×C4), (C2×C4).9D12, (C2×C8).46D6, (C2×D24).6C2, (C2×C12).99D4, (C2×C6).8SD16, C4.18(D6⋊C4), C32(M5(2)⋊C2), C8.37(C3⋊D4), (C3×M5(2))⋊9C2, C24.C411C2, (C2×C24).50C22, C2.9(C2.D24), C6.17(D4⋊C4), C12.42(C22⋊C4), C22.6(C24⋊C2), SmallGroup(192,75)

Series: Derived Chief Lower central Upper central

C1C24 — M5(2)⋊S3
C1C3C6C12C24C2×C24C2×D24 — M5(2)⋊S3
C3C6C12C24 — M5(2)⋊S3
C1C2C2×C4C2×C8M5(2)

Generators and relations for M5(2)⋊S3
 G = < a,b,c,d | a16=b2=c3=d2=1, bab=a9, ac=ca, dad=a11b, bc=cb, bd=db, dcd=c-1 >

Subgroups: 296 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6, C6, C8 [×2], C8, C2×C4, D4 [×3], C23, C12 [×2], D6 [×4], C2×C6, C16, C2×C8, M4(2), D8 [×3], C2×D4, C3⋊C8, C24 [×2], D12 [×3], C2×C12, C22×S3, C8.C4, M5(2), C2×D8, C48, D24 [×2], D24, C4.Dic3, C2×C24, C2×D12, M5(2)⋊C2, C24.C4, C3×M5(2), C2×D24, M5(2)⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, C24⋊C2, D24, D6⋊C4, M5(2)⋊C2, C2.D24, M5(2)⋊S3

Smallest permutation representation of M5(2)⋊S3
On 48 points
Generators in S48
(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)
(1 48 17)(2 33 18)(3 34 19)(4 35 20)(5 36 21)(6 37 22)(7 38 23)(8 39 24)(9 40 25)(10 41 26)(11 42 27)(12 43 28)(13 44 29)(14 45 30)(15 46 31)(16 47 32)
(2 4)(3 15)(5 13)(6 16)(7 11)(8 14)(10 12)(17 48)(18 35)(19 46)(20 33)(21 44)(22 47)(23 42)(24 45)(25 40)(26 43)(27 38)(28 41)(29 36)(30 39)(31 34)(32 37)

G:=sub<Sym(48)| (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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (1,48,17)(2,33,18)(3,34,19)(4,35,20)(5,36,21)(6,37,22)(7,38,23)(8,39,24)(9,40,25)(10,41,26)(11,42,27)(12,43,28)(13,44,29)(14,45,30)(15,46,31)(16,47,32), (2,4)(3,15)(5,13)(6,16)(7,11)(8,14)(10,12)(17,48)(18,35)(19,46)(20,33)(21,44)(22,47)(23,42)(24,45)(25,40)(26,43)(27,38)(28,41)(29,36)(30,39)(31,34)(32,37)>;

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), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (1,48,17)(2,33,18)(3,34,19)(4,35,20)(5,36,21)(6,37,22)(7,38,23)(8,39,24)(9,40,25)(10,41,26)(11,42,27)(12,43,28)(13,44,29)(14,45,30)(15,46,31)(16,47,32), (2,4)(3,15)(5,13)(6,16)(7,11)(8,14)(10,12)(17,48)(18,35)(19,46)(20,33)(21,44)(22,47)(23,42)(24,45)(25,40)(26,43)(27,38)(28,41)(29,36)(30,39)(31,34)(32,37) );

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)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47)], [(1,48,17),(2,33,18),(3,34,19),(4,35,20),(5,36,21),(6,37,22),(7,38,23),(8,39,24),(9,40,25),(10,41,26),(11,42,27),(12,43,28),(13,44,29),(14,45,30),(15,46,31),(16,47,32)], [(2,4),(3,15),(5,13),(6,16),(7,11),(8,14),(10,12),(17,48),(18,35),(19,46),(20,33),(21,44),(22,47),(23,42),(24,45),(25,40),(26,43),(27,38),(28,41),(29,36),(30,39),(31,34),(32,37)])

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B6A6B8A8B8C8D8E12A12B12C16A16B16C16D24A24B24C24D24E24F48A···48H
order1222234466888881212121616161624242424242448···48
size112242422224224242422444442222444···4

36 irreducible representations

dim111112222222222244
type+++++++++++++
imageC1C2C2C2C4S3D4D4D6D8SD16C4×S3C3⋊D4D12D24C24⋊C2M5(2)⋊C2M5(2)⋊S3
kernelM5(2)⋊S3C24.C4C3×M5(2)C2×D24D24M5(2)C24C2×C12C2×C8C12C2×C6C8C8C2×C4C4C22C3C1
# reps111141111222224424

Matrix representation of M5(2)⋊S3 in GL4(𝔽97) generated by

0010
0001
957900
181600
,
1000
0100
00960
00096
,
969600
1000
009696
0010
,
1000
969600
00218
001695
G:=sub<GL(4,GF(97))| [0,0,95,18,0,0,79,16,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,96,0,0,0,0,96],[96,1,0,0,96,0,0,0,0,0,96,1,0,0,96,0],[1,96,0,0,0,96,0,0,0,0,2,16,0,0,18,95] >;

M5(2)⋊S3 in GAP, Magma, Sage, TeX

M_5(2)\rtimes S_3
% in TeX

G:=Group("M5(2):S3");
// GroupNames label

G:=SmallGroup(192,75);
// by ID

G=gap.SmallGroup(192,75);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,422,387,268,570,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^3=d^2=1,b*a*b=a^9,a*c=c*a,d*a*d=a^11*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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