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G = C3×M6(2)  order 192 = 26·3

Direct product of C3 and M6(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C3×M6(2), C4.C48, C967C2, C323C6, C8.3C24, C48.6C4, C24.9C8, C22.C48, C12.4C16, C16.2C12, C48.30C22, (C2×C4).5C24, C16.7(C2×C6), (C2×C16).8C6, C2.3(C2×C48), (C2×C6).1C16, (C2×C48).18C2, (C2×C12).14C8, C6.13(C2×C16), (C2×C8).13C12, C4.13(C2×C24), C8.22(C2×C12), (C2×C24).32C4, C24.91(C2×C4), C12.53(C2×C8), SmallGroup(192,176)

Series: Derived Chief Lower central Upper central

C1C2 — C3×M6(2)
C1C2C4C8C16C48C96 — C3×M6(2)
C1C2 — C3×M6(2)
C1C48 — C3×M6(2)

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

2C2
2C6

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

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

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

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

120 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D8A8B8C8D8E8F12A12B12C12D12E12F16A···16H16I16J16K16L24A···24H24I24J24K24L32A···32P48A···48P48Q···48X96A···96AF
order12233444666688888812121212121216···161616161624···242424242432···3248···4848···4896···96
size1121111211221111221111221···122221···122222···21···12···22···2

120 irreducible representations

dim11111111111111111122
type+++
imageC1C2C2C3C4C4C6C6C8C8C12C12C16C16C24C24C48C48M6(2)C3×M6(2)
kernelC3×M6(2)C96C2×C48M6(2)C48C2×C24C32C2×C16C24C2×C12C16C2×C8C12C2×C6C8C2×C4C4C22C3C1
# reps12122242444488881616816

Matrix representation of C3×M6(2) in GL3(𝔽97) generated by

3500
010
001
,
9600
03342
03664
,
9600
0179
0096
G:=sub<GL(3,GF(97))| [35,0,0,0,1,0,0,0,1],[96,0,0,0,33,36,0,42,64],[96,0,0,0,1,0,0,79,96] >;

C3×M6(2) in GAP, Magma, Sage, TeX

C_3\times M_6(2)
% in TeX

G:=Group("C3xM6(2)");
// GroupNames label

G:=SmallGroup(192,176);
// by ID

G=gap.SmallGroup(192,176);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,1373,80,102,124]);
// Polycyclic

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

Export

Subgroup lattice of C3×M6(2) in TeX

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