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G = C3⋊M6(2)  order 192 = 26·3

The semidirect product of C3 and M6(2) acting via M6(2)/C2×C16=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C48.5C4, C24.6C8, C32M6(2), C12.1C16, C16.22D6, C16.2Dic3, C48.27C22, C3⋊C325C2, C4.(C3⋊C16), C8.3(C3⋊C8), (C2×C6).3C16, (C2×C12).9C8, C6.9(C2×C16), (C2×C16).7S3, C22.(C3⋊C16), (C2×C24).26C4, C12.43(C2×C8), (C2×C48).14C2, C24.81(C2×C4), (C2×C8).14Dic3, C8.22(C2×Dic3), C4.15(C2×C3⋊C8), C2.4(C2×C3⋊C16), (C2×C4).5(C3⋊C8), SmallGroup(192,58)

Series: Derived Chief Lower central Upper central

C1C6 — C3⋊M6(2)
C1C3C6C12C24C48C3⋊C32 — C3⋊M6(2)
C3C6 — C3⋊M6(2)
C1C16C2×C16

Generators and relations for C3⋊M6(2)
 G = < a,b,c | a3=b32=c2=1, bab-1=a-1, ac=ca, cbc=b17 >

2C2
2C6
3C32
3C32
3M6(2)

Smallest permutation representation of C3⋊M6(2)
On 96 points
Generators in S96
(1 72 49)(2 50 73)(3 74 51)(4 52 75)(5 76 53)(6 54 77)(7 78 55)(8 56 79)(9 80 57)(10 58 81)(11 82 59)(12 60 83)(13 84 61)(14 62 85)(15 86 63)(16 64 87)(17 88 33)(18 34 89)(19 90 35)(20 36 91)(21 92 37)(22 38 93)(23 94 39)(24 40 95)(25 96 41)(26 42 65)(27 66 43)(28 44 67)(29 68 45)(30 46 69)(31 70 47)(32 48 71)
(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)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)(34 50)(36 52)(38 54)(40 56)(42 58)(44 60)(46 62)(48 64)(65 81)(67 83)(69 85)(71 87)(73 89)(75 91)(77 93)(79 95)

G:=sub<Sym(96)| (1,72,49)(2,50,73)(3,74,51)(4,52,75)(5,76,53)(6,54,77)(7,78,55)(8,56,79)(9,80,57)(10,58,81)(11,82,59)(12,60,83)(13,84,61)(14,62,85)(15,86,63)(16,64,87)(17,88,33)(18,34,89)(19,90,35)(20,36,91)(21,92,37)(22,38,93)(23,94,39)(24,40,95)(25,96,41)(26,42,65)(27,66,43)(28,44,67)(29,68,45)(30,46,69)(31,70,47)(32,48,71), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(34,50)(36,52)(38,54)(40,56)(42,58)(44,60)(46,62)(48,64)(65,81)(67,83)(69,85)(71,87)(73,89)(75,91)(77,93)(79,95)>;

G:=Group( (1,72,49)(2,50,73)(3,74,51)(4,52,75)(5,76,53)(6,54,77)(7,78,55)(8,56,79)(9,80,57)(10,58,81)(11,82,59)(12,60,83)(13,84,61)(14,62,85)(15,86,63)(16,64,87)(17,88,33)(18,34,89)(19,90,35)(20,36,91)(21,92,37)(22,38,93)(23,94,39)(24,40,95)(25,96,41)(26,42,65)(27,66,43)(28,44,67)(29,68,45)(30,46,69)(31,70,47)(32,48,71), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (2,18)(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(34,50)(36,52)(38,54)(40,56)(42,58)(44,60)(46,62)(48,64)(65,81)(67,83)(69,85)(71,87)(73,89)(75,91)(77,93)(79,95) );

G=PermutationGroup([(1,72,49),(2,50,73),(3,74,51),(4,52,75),(5,76,53),(6,54,77),(7,78,55),(8,56,79),(9,80,57),(10,58,81),(11,82,59),(12,60,83),(13,84,61),(14,62,85),(15,86,63),(16,64,87),(17,88,33),(18,34,89),(19,90,35),(20,36,91),(21,92,37),(22,38,93),(23,94,39),(24,40,95),(25,96,41),(26,42,65),(27,66,43),(28,44,67),(29,68,45),(30,46,69),(31,70,47),(32,48,71)], [(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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(2,18),(4,20),(6,22),(8,24),(10,26),(12,28),(14,30),(16,32),(34,50),(36,52),(38,54),(40,56),(42,58),(44,60),(46,62),(48,64),(65,81),(67,83),(69,85),(71,87),(73,89),(75,91),(77,93),(79,95)])

72 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B6C8A8B8C8D8E8F12A12B12C12D16A···16H16I16J16K16L24A···24H32A···32P48A···48P
order12234446668888881212121216···161616161624···2432···3248···48
size112211222211112222221···122222···26···62···2

72 irreducible representations

dim1111111112222222222
type++++-+-
imageC1C2C2C4C4C8C8C16C16S3Dic3D6Dic3C3⋊C8C3⋊C8C3⋊C16C3⋊C16M6(2)C3⋊M6(2)
kernelC3⋊M6(2)C3⋊C32C2×C48C48C2×C24C24C2×C12C12C2×C6C2×C16C16C16C2×C8C8C2×C4C4C22C3C1
# reps12122448811112244816

Matrix representation of C3⋊M6(2) in GL4(𝔽7) generated by

1104
3150
1356
2145
,
2460
1334
4004
6032
,
5104
3550
1326
2142
G:=sub<GL(4,GF(7))| [1,3,1,2,1,1,3,1,0,5,5,4,4,0,6,5],[2,1,4,6,4,3,0,0,6,3,0,3,0,4,4,2],[5,3,1,2,1,5,3,1,0,5,2,4,4,0,6,2] >;

C3⋊M6(2) in GAP, Magma, Sage, TeX

C_3\rtimes M_6(2)
% in TeX

G:=Group("C3:M6(2)");
// GroupNames label

G:=SmallGroup(192,58);
// by ID

G=gap.SmallGroup(192,58);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,477,58,80,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^3=b^32=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^17>;
// generators/relations

Export

Subgroup lattice of C3⋊M6(2) in TeX

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