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G = C3×C6.C42order 288 = 25·32

Direct product of C3 and C6.C42

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C6.C42
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C2×C62 — Dic3×C2×C6 — C3×C6.C42
 Lower central C3 — C6 — C3×C6.C42
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C3×C6.C42
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=c, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 346 in 179 conjugacy classes, 90 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×6], C22 [×3], C22 [×4], C6 [×6], C6 [×8], C6 [×7], C2×C4 [×2], C2×C4 [×10], C23, C32, Dic3 [×4], C12 [×12], C2×C6 [×6], C2×C6 [×8], C2×C6 [×7], C22×C4, C22×C4 [×2], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×18], C22×C6 [×2], C22×C6, C2.C42, C3×Dic3 [×4], C3×C12 [×2], C62 [×3], C62 [×4], C22×Dic3 [×2], C22×C12 [×2], C22×C12 [×3], C6×Dic3 [×4], C6×Dic3 [×4], C6×C12 [×2], C6×C12 [×2], C2×C62, C6.C42, C3×C2.C42, Dic3×C2×C6 [×2], C2×C6×C12, C3×C6.C42
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], C12 [×6], D6, C2×C6, C42, C22⋊C4 [×3], C4⋊C4 [×3], C3×S3, Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C2.C42, C3×Dic3 [×2], S3×C6, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C3×Dic6, S3×C12 [×2], C3×D12, C6×Dic3, C3×C3⋊D4 [×2], C6.C42, C3×C2.C42, Dic3×C12, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C3×D6⋊C4 [×2], C3×C6.D4, C3×C6.C42

Smallest permutation representation of C3×C6.C42
On 96 points
Generators in S96
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 73)(22 74)(23 75)(24 76)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 85)(48 86)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 61)(57 62)(58 63)(59 64)(60 65)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 49)(9 50)(10 51)(11 52)(12 53)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 37)(23 38)(24 39)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 61)(36 62)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 85)
(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)
(1 37 65 80)(2 15 66 86)(3 47 67 78)(4 13 68 96)(5 45 69 76)(6 23 70 94)(7 43 71 74)(8 21 72 92)(9 41 61 84)(10 19 62 90)(11 39 63 82)(12 17 64 88)(14 35 85 50)(16 33 87 60)(18 31 89 58)(20 29 91 56)(22 27 93 54)(24 25 95 52)(26 75 53 44)(28 73 55 42)(30 83 57 40)(32 81 59 38)(34 79 49 48)(36 77 51 46)

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

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

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

108 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E ··· 4L 6A ··· 6N 6O ··· 6AI 12A ··· 12AF 12AG ··· 12AV order 1 2 ··· 2 3 3 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 2 2 2 2 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 D4 Q8 Dic3 D6 C3×S3 Dic6 C4×S3 D12 C3⋊D4 C3×D4 C3×Q8 C3×Dic3 S3×C6 C3×Dic6 S3×C12 C3×D12 C3×C3⋊D4 kernel C3×C6.C42 Dic3×C2×C6 C2×C6×C12 C6.C42 C6×Dic3 C6×C12 C22×Dic3 C22×C12 C2×Dic3 C2×C12 C22×C12 C62 C62 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C6 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 C22 # reps 1 2 1 2 8 4 4 2 16 8 1 3 1 2 1 2 2 4 2 4 6 2 4 2 4 8 4 8

Matrix representation of C3×C6.C42 in GL5(𝔽13)

 9 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 6 0 0 0 0 0 2 0 0 0 0 0 7 0 0 0 0 12 2
,
 5 0 0 0 0 0 0 2 0 0 0 6 0 0 0 0 0 0 2 10 0 0 0 6 11

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,6,0,0,0,0,0,2,0,0,0,0,0,7,12,0,0,0,0,2],[5,0,0,0,0,0,0,6,0,0,0,2,0,0,0,0,0,0,2,6,0,0,0,10,11] >;

C3×C6.C42 in GAP, Magma, Sage, TeX

C_3\times C_6.C_4^2
% in TeX

G:=Group("C3xC6.C4^2");
// GroupNames label

G:=SmallGroup(288,265);
// by ID

G=gap.SmallGroup(288,265);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,701,176,9414]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^6=c,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^5>;
// generators/relations

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