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## G = C9×C42⋊C3order 432 = 24·33

### Direct product of C9 and C42⋊C3

Aliases: C9×C42⋊C3, (C4×C36)⋊1C3, C42⋊C93C3, C421(C3×C9), C22.(C9×A4), (C2×C18).2A4, (C4×C12).1C32, (C2×C6).6(C3×A4), C3.1(C3×C42⋊C3), (C3×C42⋊C3).2C3, SmallGroup(432,99)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C9×C42⋊C3
G = < a,b,c,d | a9=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

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

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

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

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

72 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 9A ··· 9F 9G ··· 9R 12A ··· 12H 18A ··· 18F 36A ··· 36X order 1 2 3 3 3 ··· 3 4 4 4 4 6 6 9 ··· 9 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 3 1 1 16 ··· 16 3 3 3 3 3 3 1 ··· 1 16 ··· 16 3 ··· 3 3 ··· 3 3 ··· 3

72 irreducible representations

 dim 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C3 C3 C3 C9 A4 C3×A4 C42⋊C3 C9×A4 C3×C42⋊C3 C9×C42⋊C3 kernel C9×C42⋊C3 C42⋊C9 C4×C36 C3×C42⋊C3 C42⋊C3 C2×C18 C2×C6 C9 C22 C3 C1 # reps 1 4 2 2 18 1 2 4 6 8 24

Matrix representation of C9×C42⋊C3 in GL6(𝔽37)

 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 31 0 0 0 0 0 0 31 0 0 0 0 0 0 36
,
 36 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 31
,
 0 26 0 0 0 0 0 0 26 0 0 0 26 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

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

C9×C42⋊C3 in GAP, Magma, Sage, TeX

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

G:=Group("C9xC4^2:C3");
// GroupNames label

G:=SmallGroup(432,99);
// by ID

G=gap.SmallGroup(432,99);
# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,50,1515,360,10399,102,9077,15882]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^4=c^4=d^3=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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