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## G = C3×C42⋊2S3order 288 = 25·32

### Direct product of C3 and C42⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C42⋊2S3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×C42⋊2S3
 Lower central C3 — C6 — C3×C42⋊2S3
 Upper central C1 — C2×C12 — C4×C12

Generators and relations for C3×C422S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 338 in 167 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×2], C2×C4 [×7], C23, C32, Dic3 [×2], Dic3 [×2], C12 [×4], C12 [×14], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3×S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×10], C22×S3, C22×C6, C42⋊C2, C3×Dic3 [×2], C3×Dic3 [×2], C3×C12 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×2], C4×C12 [×2], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4, C22×C12, S3×C12 [×4], C6×Dic3, C6×Dic3 [×2], C6×C12, C6×C12 [×2], S3×C2×C6, C422S3, C3×C42⋊C2, Dic3×C12, C3×Dic3⋊C4 [×2], C3×D6⋊C4 [×2], C122, S3×C2×C12, C3×C422S3
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C4○D4 [×2], C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, C42⋊C2, S3×C6 [×3], S3×C2×C4, C4○D12 [×2], C22×C12, C3×C4○D4 [×2], S3×C12 [×2], S3×C2×C6, C422S3, C3×C42⋊C2, S3×C2×C12, C3×C4○D12 [×2], C3×C422S3

Smallest permutation representation of C3×C422S3
On 96 points
Generators in S96
(1 9 31)(2 10 32)(3 11 29)(4 12 30)(5 36 22)(6 33 23)(7 34 24)(8 35 21)(13 19 41)(14 20 42)(15 17 43)(16 18 44)(25 63 95)(26 64 96)(27 61 93)(28 62 94)(37 47 49)(38 48 50)(39 45 51)(40 46 52)(53 87 57)(54 88 58)(55 85 59)(56 86 60)(65 91 77)(66 92 78)(67 89 79)(68 90 80)(69 73 83)(70 74 84)(71 75 81)(72 76 82)
(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 39 15 23)(2 40 16 24)(3 37 13 21)(4 38 14 22)(5 12 48 20)(6 9 45 17)(7 10 46 18)(8 11 47 19)(25 91 83 53)(26 92 84 54)(27 89 81 55)(28 90 82 56)(29 49 41 35)(30 50 42 36)(31 51 43 33)(32 52 44 34)(57 95 65 73)(58 96 66 74)(59 93 67 75)(60 94 68 76)(61 79 71 85)(62 80 72 86)(63 77 69 87)(64 78 70 88)
(1 31 9)(2 32 10)(3 29 11)(4 30 12)(5 22 36)(6 23 33)(7 24 34)(8 21 35)(13 41 19)(14 42 20)(15 43 17)(16 44 18)(25 63 95)(26 64 96)(27 61 93)(28 62 94)(37 49 47)(38 50 48)(39 51 45)(40 52 46)(53 87 57)(54 88 58)(55 85 59)(56 86 60)(65 91 77)(66 92 78)(67 89 79)(68 90 80)(69 73 83)(70 74 84)(71 75 81)(72 76 82)
(1 87)(2 88)(3 85)(4 86)(5 74)(6 75)(7 76)(8 73)(9 57)(10 58)(11 59)(12 60)(13 79)(14 80)(15 77)(16 78)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 49)(26 50)(27 51)(28 52)(29 55)(30 56)(31 53)(32 54)(33 81)(34 82)(35 83)(36 84)(37 63)(38 64)(39 61)(40 62)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 12A ··· 12H 12I ··· 12AZ 12BA ··· 12BL order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 6 6 1 1 2 2 2 1 1 1 1 2 2 2 2 6 ··· 6 1 ··· 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 S3 D6 C4○D4 C3×S3 C4×S3 S3×C6 C4○D12 C3×C4○D4 S3×C12 C3×C4○D12 kernel C3×C42⋊2S3 Dic3×C12 C3×Dic3⋊C4 C3×D6⋊C4 C122 S3×C2×C12 C42⋊2S3 S3×C12 C4×Dic3 Dic3⋊C4 D6⋊C4 C4×C12 S3×C2×C4 C4×S3 C4×C12 C2×C12 C3×C6 C42 C12 C2×C4 C6 C6 C4 C2 # reps 1 1 2 2 1 1 2 8 2 4 4 2 2 16 1 3 4 2 4 6 8 8 8 16

Matrix representation of C3×C422S3 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 12 0 0 0 0 12 0 0 0 0 8 0 0 0 0 8
,
 5 0 0 0 0 5 0 0 0 0 1 0 0 0 0 12
,
 3 11 0 0 0 9 0 0 0 0 3 0 0 0 0 9
,
 6 7 0 0 8 7 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,12,0,0,0,0,8,0,0,0,0,8],[5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,12],[3,0,0,0,11,9,0,0,0,0,3,0,0,0,0,9],[6,8,0,0,7,7,0,0,0,0,0,1,0,0,1,0] >;

C3×C422S3 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes_2S_3
% in TeX

G:=Group("C3xC4^2:2S3");
// GroupNames label

G:=SmallGroup(288,643);
// by ID

G=gap.SmallGroup(288,643);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,1094,142,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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