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## G = C9×C22⋊A4order 432 = 24·33

### Direct product of C9 and C22⋊A4

Aliases: C9×C22⋊A4, (C2×C18)⋊1A4, C244(C3×C9), C222(C9×A4), (C23×C18)⋊2C3, C24⋊C97C3, (C23×C6).11C32, (C2×C6).12(C3×A4), C3.1(C3×C22⋊A4), (C3×C22⋊A4).4C3, SmallGroup(432,551)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C9×C22⋊A4
 Chief series C1 — C22 — C24 — C23×C6 — C3×C22⋊A4 — C9×C22⋊A4
 Lower central C24 — C9×C22⋊A4
 Upper central C1 — C9

Generators and relations for C9×C22⋊A4
G = < a,b,c,d,e,f | a9=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >

Subgroups: 460 in 130 conjugacy classes, 28 normal (8 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C9, C9, C32, A4, C2×C6, C2×C6, C24, C18, C22×C6, C3×C9, C3.A4, C2×C18, C2×C18, C3×A4, C22⋊A4, C23×C6, C22×C18, C9×A4, C24⋊C9, C23×C18, C3×C22⋊A4, C9×C22⋊A4
Quotients: C1, C3, C9, C32, A4, C3×C9, C3×A4, C22⋊A4, C9×A4, C3×C22⋊A4, C9×C22⋊A4

Smallest permutation representation of C9×C22⋊A4
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 42)(2 43)(3 44)(4 45)(5 37)(6 38)(7 39)(8 40)(9 41)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(46 62)(47 63)(48 55)(49 56)(50 57)(51 58)(52 59)(53 60)(54 61)
(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)(82 98)(83 99)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 46)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 82)(73 99)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)
(1 52)(2 53)(3 54)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 100)(37 63)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)(82 98)(83 99)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)
(1 108 66)(2 100 67)(3 101 68)(4 102 69)(5 103 70)(6 104 71)(7 105 72)(8 106 64)(9 107 65)(10 76 43)(11 77 44)(12 78 45)(13 79 37)(14 80 38)(15 81 39)(16 73 40)(17 74 41)(18 75 42)(19 85 52)(20 86 53)(21 87 54)(22 88 46)(23 89 47)(24 90 48)(25 82 49)(26 83 50)(27 84 51)(28 94 61)(29 95 62)(30 96 63)(31 97 55)(32 98 56)(33 99 57)(34 91 58)(35 92 59)(36 93 60)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2E 3A 3B 3C ··· 3H 6A ··· 6J 9A ··· 9F 9G ··· 9R 18A ··· 18AD order 1 2 ··· 2 3 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 ··· 3 1 1 16 ··· 16 3 ··· 3 1 ··· 1 16 ··· 16 3 ··· 3

72 irreducible representations

 dim 1 1 1 1 1 3 3 3 type + + image C1 C3 C3 C3 C9 A4 C3×A4 C9×A4 kernel C9×C22⋊A4 C24⋊C9 C23×C18 C3×C22⋊A4 C22⋊A4 C2×C18 C2×C6 C22 # reps 1 4 2 2 18 5 10 30

Matrix representation of C9×C22⋊A4 in GL6(𝔽19)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 18
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 18 0 0 0 0 0 0 18 0 0 0 0 6 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 18 0 0 0 0 0 9 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1
,
 15 16 0 0 0 0 13 4 11 0 0 0 16 9 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 7 0 0

G:=sub<GL(6,GF(19))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[18,0,6,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[18,9,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1],[15,13,16,0,0,0,16,4,9,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,7,0,0,0,0,0,0,7,0] >;

C9×C22⋊A4 in GAP, Magma, Sage, TeX

C_9\times C_2^2\rtimes A_4
% in TeX

G:=Group("C9xC2^2:A4");
// GroupNames label

G:=SmallGroup(432,551);
// by ID

G=gap.SmallGroup(432,551);
# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,50,1515,2839,9077,15882]);
// Polycyclic

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

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