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## G = C3×C42⋊C9order 432 = 24·33

### Direct product of C3 and C42⋊C9

Aliases: C3×C42⋊C9, C62.1A4, C122.1C3, (C4×C12)⋊C9, C422(C3×C9), (C4×C12).3C32, C32.2(C42⋊C3), (C2×C6).8(C3×A4), C3.2(C3×C42⋊C3), C22.(C3×C3.A4), (C2×C6).2(C3.A4), SmallGroup(432,101)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C3×C42⋊C9
 Chief series C1 — C22 — C42 — C4×C12 — C42⋊C9 — C3×C42⋊C9
 Lower central C42 — C3×C42⋊C9
 Upper central C1 — C32

Generators and relations for C3×C42⋊C9
G = < a,b,c,d | a3=b4=c4=d9=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

Smallest permutation representation of C3×C42⋊C9
On 108 points
Generators in S108
(1 20 92)(2 21 93)(3 22 94)(4 23 95)(5 24 96)(6 25 97)(7 26 98)(8 27 99)(9 19 91)(10 43 87)(11 44 88)(12 45 89)(13 37 90)(14 38 82)(15 39 83)(16 40 84)(17 41 85)(18 42 86)(28 106 48)(29 107 49)(30 108 50)(31 100 51)(32 101 52)(33 102 53)(34 103 54)(35 104 46)(36 105 47)(55 79 67)(56 80 68)(57 81 69)(58 73 70)(59 74 71)(60 75 72)(61 76 64)(62 77 65)(63 78 66)
(1 17 52 59)(2 53)(3 10 54 61)(4 11 46 62)(5 47)(6 13 48 55)(7 14 49 56)(8 50)(9 16 51 58)(12 63)(15 57)(18 60)(19 40 31 73)(20 41 32 74)(21 33)(22 43 34 76)(23 44 35 77)(24 36)(25 37 28 79)(26 38 29 80)(27 30)(39 81)(42 75)(45 78)(64 94 87 103)(65 95 88 104)(66 89)(67 97 90 106)(68 98 82 107)(69 83)(70 91 84 100)(71 92 85 101)(72 86)(93 102)(96 105)(99 108)
(1 59 52 17)(2 18 53 60)(4 62 46 11)(5 12 47 63)(7 56 49 14)(8 15 50 57)(20 74 32 41)(21 42 33 75)(23 77 35 44)(24 45 36 78)(26 80 29 38)(27 39 30 81)(65 104 88 95)(66 96 89 105)(68 107 82 98)(69 99 83 108)(71 101 85 92)(72 93 86 102)
(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 97 98 99)(100 101 102 103 104 105 106 107 108)

G:=sub<Sym(108)| (1,20,92)(2,21,93)(3,22,94)(4,23,95)(5,24,96)(6,25,97)(7,26,98)(8,27,99)(9,19,91)(10,43,87)(11,44,88)(12,45,89)(13,37,90)(14,38,82)(15,39,83)(16,40,84)(17,41,85)(18,42,86)(28,106,48)(29,107,49)(30,108,50)(31,100,51)(32,101,52)(33,102,53)(34,103,54)(35,104,46)(36,105,47)(55,79,67)(56,80,68)(57,81,69)(58,73,70)(59,74,71)(60,75,72)(61,76,64)(62,77,65)(63,78,66), (1,17,52,59)(2,53)(3,10,54,61)(4,11,46,62)(5,47)(6,13,48,55)(7,14,49,56)(8,50)(9,16,51,58)(12,63)(15,57)(18,60)(19,40,31,73)(20,41,32,74)(21,33)(22,43,34,76)(23,44,35,77)(24,36)(25,37,28,79)(26,38,29,80)(27,30)(39,81)(42,75)(45,78)(64,94,87,103)(65,95,88,104)(66,89)(67,97,90,106)(68,98,82,107)(69,83)(70,91,84,100)(71,92,85,101)(72,86)(93,102)(96,105)(99,108), (1,59,52,17)(2,18,53,60)(4,62,46,11)(5,12,47,63)(7,56,49,14)(8,15,50,57)(20,74,32,41)(21,42,33,75)(23,77,35,44)(24,45,36,78)(26,80,29,38)(27,39,30,81)(65,104,88,95)(66,96,89,105)(68,107,82,98)(69,99,83,108)(71,101,85,92)(72,93,86,102), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)>;

G:=Group( (1,20,92)(2,21,93)(3,22,94)(4,23,95)(5,24,96)(6,25,97)(7,26,98)(8,27,99)(9,19,91)(10,43,87)(11,44,88)(12,45,89)(13,37,90)(14,38,82)(15,39,83)(16,40,84)(17,41,85)(18,42,86)(28,106,48)(29,107,49)(30,108,50)(31,100,51)(32,101,52)(33,102,53)(34,103,54)(35,104,46)(36,105,47)(55,79,67)(56,80,68)(57,81,69)(58,73,70)(59,74,71)(60,75,72)(61,76,64)(62,77,65)(63,78,66), (1,17,52,59)(2,53)(3,10,54,61)(4,11,46,62)(5,47)(6,13,48,55)(7,14,49,56)(8,50)(9,16,51,58)(12,63)(15,57)(18,60)(19,40,31,73)(20,41,32,74)(21,33)(22,43,34,76)(23,44,35,77)(24,36)(25,37,28,79)(26,38,29,80)(27,30)(39,81)(42,75)(45,78)(64,94,87,103)(65,95,88,104)(66,89)(67,97,90,106)(68,98,82,107)(69,83)(70,91,84,100)(71,92,85,101)(72,86)(93,102)(96,105)(99,108), (1,59,52,17)(2,18,53,60)(4,62,46,11)(5,12,47,63)(7,56,49,14)(8,15,50,57)(20,74,32,41)(21,42,33,75)(23,77,35,44)(24,45,36,78)(26,80,29,38)(27,39,30,81)(65,104,88,95)(66,96,89,105)(68,107,82,98)(69,99,83,108)(71,101,85,92)(72,93,86,102), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108) );

G=PermutationGroup([[(1,20,92),(2,21,93),(3,22,94),(4,23,95),(5,24,96),(6,25,97),(7,26,98),(8,27,99),(9,19,91),(10,43,87),(11,44,88),(12,45,89),(13,37,90),(14,38,82),(15,39,83),(16,40,84),(17,41,85),(18,42,86),(28,106,48),(29,107,49),(30,108,50),(31,100,51),(32,101,52),(33,102,53),(34,103,54),(35,104,46),(36,105,47),(55,79,67),(56,80,68),(57,81,69),(58,73,70),(59,74,71),(60,75,72),(61,76,64),(62,77,65),(63,78,66)], [(1,17,52,59),(2,53),(3,10,54,61),(4,11,46,62),(5,47),(6,13,48,55),(7,14,49,56),(8,50),(9,16,51,58),(12,63),(15,57),(18,60),(19,40,31,73),(20,41,32,74),(21,33),(22,43,34,76),(23,44,35,77),(24,36),(25,37,28,79),(26,38,29,80),(27,30),(39,81),(42,75),(45,78),(64,94,87,103),(65,95,88,104),(66,89),(67,97,90,106),(68,98,82,107),(69,83),(70,91,84,100),(71,92,85,101),(72,86),(93,102),(96,105),(99,108)], [(1,59,52,17),(2,18,53,60),(4,62,46,11),(5,12,47,63),(7,56,49,14),(8,15,50,57),(20,74,32,41),(21,42,33,75),(23,77,35,44),(24,45,36,78),(26,80,29,38),(27,39,30,81),(65,104,88,95),(66,96,89,105),(68,107,82,98),(69,99,83,108),(71,101,85,92),(72,93,86,102)], [(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),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)]])

72 conjugacy classes

 class 1 2 3A ··· 3H 4A 4B 4C 4D 6A ··· 6H 9A ··· 9R 12A ··· 12AF order 1 2 3 ··· 3 4 4 4 4 6 ··· 6 9 ··· 9 12 ··· 12 size 1 3 1 ··· 1 3 3 3 3 3 ··· 3 16 ··· 16 3 ··· 3

72 irreducible representations

 dim 1 1 1 1 3 3 3 3 3 3 type + + image C1 C3 C3 C9 A4 C3.A4 C3×A4 C42⋊C3 C42⋊C9 C3×C42⋊C3 kernel C3×C42⋊C9 C42⋊C9 C122 C4×C12 C62 C2×C6 C2×C6 C32 C3 C3 # reps 1 6 2 18 1 6 2 4 24 8

Matrix representation of C3×C42⋊C9 in GL7(𝔽37)

 26 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 31
,
 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0

G:=sub<GL(7,GF(37))| [26,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,31],[10,0,0,0,0,0,0,0,0,0,26,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

C3×C42⋊C9 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes C_9
% in TeX

G:=Group("C3xC4^2:C9");
// GroupNames label

G:=SmallGroup(432,101);
// by ID

G=gap.SmallGroup(432,101);
# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,63,1515,360,10399,102,9077,15882]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^4=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations

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