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## G = C3×C42⋊3S3order 288 = 25·32

### Direct product of C3 and C42⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C42⋊3S3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C3×D6⋊C4 — C3×C42⋊3S3
 Lower central C3 — C2×C6 — C3×C42⋊3S3
 Upper central C1 — C2×C6 — C4×C12

Generators and relations for C3×C423S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc2, cd=dc, ece=b2c-1, ede=d-1 >

Subgroups: 306 in 135 conjugacy classes, 58 normal (16 characteristic)
C1, C2 [×3], C2, C3 [×2], C3, C4 [×6], C22, C22 [×3], S3, C6 [×6], C6 [×4], C2×C4 [×3], C2×C4 [×3], C23, C32, Dic3 [×3], C12 [×15], D6 [×3], C2×C6 [×2], C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3×S3, C3×C6 [×3], C2×Dic3 [×3], C2×C12 [×6], C2×C12 [×6], C22×S3, C22×C6, C422C2, C3×Dic3 [×3], C3×C12 [×3], S3×C6 [×3], C62, Dic3⋊C4 [×3], D6⋊C4 [×3], C4×C12 [×2], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C6×Dic3 [×3], C6×C12 [×3], S3×C2×C6, C423S3, C3×C422C2, C3×Dic3⋊C4 [×3], C3×D6⋊C4 [×3], C122, C3×C423S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C4○D4 [×3], C3×S3, C22×S3, C22×C6, C422C2, S3×C6 [×3], C4○D12 [×3], C3×C4○D4 [×3], S3×C2×C6, C423S3, C3×C422C2, C3×C4○D12 [×3], C3×C423S3

Smallest permutation representation of C3×C423S3
On 96 points
Generators in S96
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 68 31)(6 65 32)(7 66 29)(8 67 30)(13 43 54)(14 44 55)(15 41 56)(16 42 53)(17 39 45)(18 40 46)(19 37 47)(20 38 48)(25 89 87)(26 90 88)(27 91 85)(28 92 86)(33 79 73)(34 80 74)(35 77 75)(36 78 76)(49 70 84)(50 71 81)(51 72 82)(52 69 83)(57 93 62)(58 94 63)(59 95 64)(60 96 61)
(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 75 19 41)(2 76 20 42)(3 73 17 43)(4 74 18 44)(5 50 87 63)(6 51 88 64)(7 52 85 61)(8 49 86 62)(9 77 47 15)(10 78 48 16)(11 79 45 13)(12 80 46 14)(21 33 39 54)(22 34 40 55)(23 35 37 56)(24 36 38 53)(25 58 68 71)(26 59 65 72)(27 60 66 69)(28 57 67 70)(29 83 91 96)(30 84 92 93)(31 81 89 94)(32 82 90 95)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 68 31)(6 65 32)(7 66 29)(8 67 30)(13 54 43)(14 55 44)(15 56 41)(16 53 42)(17 45 39)(18 46 40)(19 47 37)(20 48 38)(25 89 87)(26 90 88)(27 91 85)(28 92 86)(33 73 79)(34 74 80)(35 75 77)(36 76 78)(49 70 84)(50 71 81)(51 72 82)(52 69 83)(57 93 62)(58 94 63)(59 95 64)(60 96 61)
(1 64)(2 52)(3 62)(4 50)(5 76)(6 43)(7 74)(8 41)(9 95)(10 83)(11 93)(12 81)(13 32)(14 91)(15 30)(16 89)(17 49)(18 63)(19 51)(20 61)(21 57)(22 71)(23 59)(24 69)(25 53)(26 33)(27 55)(28 35)(29 80)(31 78)(34 66)(36 68)(37 72)(38 60)(39 70)(40 58)(42 87)(44 85)(45 84)(46 94)(47 82)(48 96)(54 65)(56 67)(73 88)(75 86)(77 92)(79 90)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A ··· 4F 4G 4H 4I 6A ··· 6F 6G ··· 6O 6P 6Q 12A ··· 12AV 12AW ··· 12BB order 1 2 2 2 2 3 3 3 3 3 4 ··· 4 4 4 4 6 ··· 6 6 ··· 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 12 1 1 2 2 2 2 ··· 2 12 12 12 1 ··· 1 2 ··· 2 12 12 2 ··· 2 12 ··· 12

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 C4○D4 C3×S3 S3×C6 C4○D12 C3×C4○D4 C3×C4○D12 kernel C3×C42⋊3S3 C3×Dic3⋊C4 C3×D6⋊C4 C122 C42⋊3S3 Dic3⋊C4 D6⋊C4 C4×C12 C4×C12 C2×C12 C3×C6 C42 C2×C4 C6 C6 C2 # reps 1 3 3 1 2 6 6 2 1 3 6 2 6 12 12 24

Matrix representation of C3×C423S3 in GL4(𝔽13) generated by

 3 0 0 0 0 3 0 0 0 0 9 0 0 0 0 9
,
 12 0 0 0 3 1 0 0 0 0 5 0 0 0 0 5
,
 8 0 0 0 2 5 0 0 0 0 12 0 0 0 0 1
,
 9 0 0 0 4 3 0 0 0 0 3 0 0 0 0 9
,
 12 8 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[12,3,0,0,0,1,0,0,0,0,5,0,0,0,0,5],[8,2,0,0,0,5,0,0,0,0,12,0,0,0,0,1],[9,4,0,0,0,3,0,0,0,0,3,0,0,0,0,9],[12,0,0,0,8,1,0,0,0,0,0,1,0,0,1,0] >;

C3×C423S3 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes_3S_3
% in TeX

G:=Group("C3xC4^2:3S3");
// GroupNames label

G:=SmallGroup(288,647);
// by ID

G=gap.SmallGroup(288,647);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,176,1598,268,9414]);
// 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,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c^-1,e*d*e=d^-1>;
// generators/relations

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