Copied to
clipboard

## 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, C2, C3, C3, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22×C4, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C2.C42, C3×Dic3, C3×C12, C62, C62, C22×Dic3, C22×C12, C22×C12, C6×Dic3, C6×Dic3, C6×C12, C6×C12, C2×C62, C6.C42, C3×C2.C42, Dic3×C2×C6, C2×C6×C12, C3×C6.C42
Quotients:

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

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

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

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

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

׿
×
𝔽