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## G = C3×C42⋊7S3order 288 = 25·32

### Direct product of C3 and C42⋊7S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×C42⋊7S3
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C3×D6⋊C4 — C3×C42⋊7S3
 Lower central C3 — C2×C6 — C3×C42⋊7S3
 Upper central C1 — C2×C6 — C4×C12

Generators and relations for C3×C427S3
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, ede=d-1 >

Subgroups: 434 in 167 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×2], C12 [×4], C12 [×12], D6 [×6], C2×C6 [×2], C2×C6 [×7], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×2], D12 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×C6 [×2], C4.4D4, C3×Dic3 [×2], C3×C12 [×2], C3×C12 [×2], S3×C6 [×6], C62, D6⋊C4 [×4], C4×C12 [×2], C4×C12, C3×C22⋊C4 [×4], C2×Dic6, C2×D12, C6×D4, C6×Q8, C3×Dic6 [×2], C3×D12 [×2], C6×Dic3 [×2], C6×C12, C6×C12 [×2], S3×C2×C6 [×2], C427S3, C3×C4.4D4, C3×D6⋊C4 [×4], C122, C6×Dic6, C6×D12, C3×C427S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C4○D4 [×2], C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C4.4D4, S3×C6 [×3], C2×D12, C4○D12 [×2], C6×D4, C3×C4○D4 [×2], C3×D12 [×2], S3×C2×C6, C427S3, C3×C4.4D4, C6×D12, C3×C4○D12 [×2], C3×C427S3

Smallest permutation representation of C3×C427S3
On 96 points
Generators in S96
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 64 68)(6 61 65)(7 62 66)(8 63 67)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(17 73 39)(18 74 40)(19 75 37)(20 76 38)(25 87 29)(26 88 30)(27 85 31)(28 86 32)(33 50 54)(34 51 55)(35 52 56)(36 49 53)(57 93 89)(58 94 90)(59 95 91)(60 96 92)(69 83 79)(70 84 80)(71 81 77)(72 82 78)
(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 54 43 17)(2 55 44 18)(3 56 41 19)(4 53 42 20)(5 69 87 60)(6 70 88 57)(7 71 85 58)(8 72 86 59)(9 50 47 39)(10 51 48 40)(11 52 45 37)(12 49 46 38)(13 73 23 33)(14 74 24 34)(15 75 21 35)(16 76 22 36)(25 92 68 79)(26 89 65 80)(27 90 66 77)(28 91 67 78)(29 96 64 83)(30 93 61 84)(31 94 62 81)(32 95 63 82)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 64 68)(6 61 65)(7 62 66)(8 63 67)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 39 73)(18 40 74)(19 37 75)(20 38 76)(25 87 29)(26 88 30)(27 85 31)(28 86 32)(33 54 50)(34 55 51)(35 56 52)(36 53 49)(57 93 89)(58 94 90)(59 95 91)(60 96 92)(69 83 79)(70 84 80)(71 81 77)(72 82 78)
(1 91)(2 79)(3 89)(4 77)(5 76)(6 33)(7 74)(8 35)(9 95)(10 83)(11 93)(12 81)(13 72)(14 60)(15 70)(16 58)(17 26)(18 66)(19 28)(20 68)(21 57)(22 71)(23 59)(24 69)(25 53)(27 55)(29 49)(30 39)(31 51)(32 37)(34 85)(36 87)(38 64)(40 62)(41 80)(42 90)(43 78)(44 92)(45 84)(46 94)(47 82)(48 96)(50 61)(52 63)(54 65)(56 67)(73 88)(75 86)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A ··· 4F 4G 4H 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 12A ··· 12AV 12AW 12AX 12AY 12AZ order 1 2 2 2 2 2 3 3 3 3 3 4 ··· 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 12 12 1 1 2 2 2 2 ··· 2 12 12 1 ··· 1 2 ··· 2 12 12 12 12 2 ··· 2 12 12 12 12

90 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 type + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D6 C4○D4 C3×S3 D12 C3×D4 S3×C6 C4○D12 C3×C4○D4 C3×D12 C3×C4○D12 kernel C3×C42⋊7S3 C3×D6⋊C4 C122 C6×Dic6 C6×D12 C42⋊7S3 D6⋊C4 C4×C12 C2×Dic6 C2×D12 C4×C12 C3×C12 C2×C12 C3×C6 C42 C12 C12 C2×C4 C6 C6 C4 C2 # reps 1 4 1 1 1 2 8 2 2 2 1 2 3 4 2 4 4 6 8 8 8 16

Matrix representation of C3×C427S3 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3
,
 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 12
,
 0 1 0 0 1 0 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 3
,
 6 3 0 0 10 7 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,12],[0,1,0,0,1,0,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[6,10,0,0,3,7,0,0,0,0,0,1,0,0,1,0] >;

C3×C427S3 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes_7S_3
% in TeX

G:=Group("C3xC4^2:7S3");
// GroupNames label

G:=SmallGroup(288,646);
// by ID

G=gap.SmallGroup(288,646);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,176,590,268,9414]);
// 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,e*d*e=d^-1>;
// generators/relations

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