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## G = C22×C32⋊C9order 324 = 22·34

### Direct product of C22 and C32⋊C9

direct product, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×C32⋊C9
 Chief series C1 — C3 — C32 — C33 — C32⋊C9 — C2×C32⋊C9 — C22×C32⋊C9
 Lower central C1 — C3 — C22×C32⋊C9
 Upper central C1 — C62 — C22×C32⋊C9

Generators and relations for C22×C32⋊C9
G = < a,b,c,d,e | a2=b2=c3=d3=e9=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 205 in 115 conjugacy classes, 70 normal (12 characteristic)
C1, C2, C3, C3, C3, C22, C6, C6, C9, C32, C32, C32, C2×C6, C2×C6, C2×C6, C18, C3×C6, C3×C6, C3×C9, C33, C2×C18, C62, C62, C62, C3×C18, C32×C6, C32⋊C9, C6×C18, C3×C62, C2×C32⋊C9, C22×C32⋊C9
Quotients: C1, C2, C3, C22, C6, C9, C32, C2×C6, C18, C3×C6, C3×C9, He3, 3- 1+2, C2×C18, C62, C3×C18, C2×He3, C2×3- 1+2, C32⋊C9, C6×C18, C22×He3, C22×3- 1+2, C2×C32⋊C9, C22×C32⋊C9

Smallest permutation representation of C22×C32⋊C9
On 108 points
Generators in S108
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 73)(9 74)(10 61)(11 62)(12 63)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 70)(20 71)(21 72)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 46)(9 47)(10 88)(11 89)(12 90)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 97)(20 98)(21 99)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)
(1 4 7)(2 28 42)(3 40 32)(5 31 45)(6 43 35)(8 34 39)(9 37 29)(10 24 100)(11 107 19)(12 15 18)(13 27 103)(14 101 22)(16 21 106)(17 104 25)(20 23 26)(30 33 36)(38 41 44)(46 61 66)(47 64 56)(48 51 54)(49 55 69)(50 67 59)(52 58 72)(53 70 62)(57 60 63)(65 68 71)(73 88 93)(74 91 83)(75 78 81)(76 82 96)(77 94 86)(79 85 99)(80 97 89)(84 87 90)(92 95 98)(102 105 108)
(1 44 33)(2 45 34)(3 37 35)(4 38 36)(5 39 28)(6 40 29)(7 41 30)(8 42 31)(9 43 32)(10 103 21)(11 104 22)(12 105 23)(13 106 24)(14 107 25)(15 108 26)(16 100 27)(17 101 19)(18 102 20)(46 69 58)(47 70 59)(48 71 60)(49 72 61)(50 64 62)(51 65 63)(52 66 55)(53 67 56)(54 68 57)(73 96 85)(74 97 86)(75 98 87)(76 99 88)(77 91 89)(78 92 90)(79 93 82)(80 94 83)(81 95 84)
(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)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,73)(9,74)(10,61)(11,62)(12,63)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,70)(20,71)(21,72)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,88)(11,89)(12,90)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,97)(20,98)(21,99)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (1,4,7)(2,28,42)(3,40,32)(5,31,45)(6,43,35)(8,34,39)(9,37,29)(10,24,100)(11,107,19)(12,15,18)(13,27,103)(14,101,22)(16,21,106)(17,104,25)(20,23,26)(30,33,36)(38,41,44)(46,61,66)(47,64,56)(48,51,54)(49,55,69)(50,67,59)(52,58,72)(53,70,62)(57,60,63)(65,68,71)(73,88,93)(74,91,83)(75,78,81)(76,82,96)(77,94,86)(79,85,99)(80,97,89)(84,87,90)(92,95,98)(102,105,108), (1,44,33)(2,45,34)(3,37,35)(4,38,36)(5,39,28)(6,40,29)(7,41,30)(8,42,31)(9,43,32)(10,103,21)(11,104,22)(12,105,23)(13,106,24)(14,107,25)(15,108,26)(16,100,27)(17,101,19)(18,102,20)(46,69,58)(47,70,59)(48,71,60)(49,72,61)(50,64,62)(51,65,63)(52,66,55)(53,67,56)(54,68,57)(73,96,85)(74,97,86)(75,98,87)(76,99,88)(77,91,89)(78,92,90)(79,93,82)(80,94,83)(81,95,84), (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( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,73)(9,74)(10,61)(11,62)(12,63)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,70)(20,71)(21,72)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,88)(11,89)(12,90)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,97)(20,98)(21,99)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (1,4,7)(2,28,42)(3,40,32)(5,31,45)(6,43,35)(8,34,39)(9,37,29)(10,24,100)(11,107,19)(12,15,18)(13,27,103)(14,101,22)(16,21,106)(17,104,25)(20,23,26)(30,33,36)(38,41,44)(46,61,66)(47,64,56)(48,51,54)(49,55,69)(50,67,59)(52,58,72)(53,70,62)(57,60,63)(65,68,71)(73,88,93)(74,91,83)(75,78,81)(76,82,96)(77,94,86)(79,85,99)(80,97,89)(84,87,90)(92,95,98)(102,105,108), (1,44,33)(2,45,34)(3,37,35)(4,38,36)(5,39,28)(6,40,29)(7,41,30)(8,42,31)(9,43,32)(10,103,21)(11,104,22)(12,105,23)(13,106,24)(14,107,25)(15,108,26)(16,100,27)(17,101,19)(18,102,20)(46,69,58)(47,70,59)(48,71,60)(49,72,61)(50,64,62)(51,65,63)(52,66,55)(53,67,56)(54,68,57)(73,96,85)(74,97,86)(75,98,87)(76,99,88)(77,91,89)(78,92,90)(79,93,82)(80,94,83)(81,95,84), (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([[(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,73),(9,74),(10,61),(11,62),(12,63),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,70),(20,71),(21,72),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,46),(9,47),(10,88),(11,89),(12,90),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,97),(20,98),(21,99),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108)], [(1,4,7),(2,28,42),(3,40,32),(5,31,45),(6,43,35),(8,34,39),(9,37,29),(10,24,100),(11,107,19),(12,15,18),(13,27,103),(14,101,22),(16,21,106),(17,104,25),(20,23,26),(30,33,36),(38,41,44),(46,61,66),(47,64,56),(48,51,54),(49,55,69),(50,67,59),(52,58,72),(53,70,62),(57,60,63),(65,68,71),(73,88,93),(74,91,83),(75,78,81),(76,82,96),(77,94,86),(79,85,99),(80,97,89),(84,87,90),(92,95,98),(102,105,108)], [(1,44,33),(2,45,34),(3,37,35),(4,38,36),(5,39,28),(6,40,29),(7,41,30),(8,42,31),(9,43,32),(10,103,21),(11,104,22),(12,105,23),(13,106,24),(14,107,25),(15,108,26),(16,100,27),(17,101,19),(18,102,20),(46,69,58),(47,70,59),(48,71,60),(49,72,61),(50,64,62),(51,65,63),(52,66,55),(53,67,56),(54,68,57),(73,96,85),(74,97,86),(75,98,87),(76,99,88),(77,91,89),(78,92,90),(79,93,82),(80,94,83),(81,95,84)], [(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)]])

132 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3N 6A ··· 6X 6Y ··· 6AP 9A ··· 9R 18A ··· 18BB order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C9 C18 He3 3- 1+2 C2×He3 C2×3- 1+2 kernel C22×C32⋊C9 C2×C32⋊C9 C6×C18 C3×C62 C3×C18 C32×C6 C62 C3×C6 C2×C6 C2×C6 C6 C6 # reps 1 3 6 2 18 6 18 54 2 4 6 12

Matrix representation of C22×C32⋊C9 in GL5(𝔽19)

 1 0 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 11 0 0 0 0 0 11 0 0 0 0 0 11 14 15 0 0 0 1 12 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 6 0 0 0 0 0 4 0 0 0 0 0 5 0 0 0 0 11 4 17 0 0 15 2 10

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,14,1,0,0,0,15,12,7],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[6,0,0,0,0,0,4,0,0,0,0,0,5,11,15,0,0,0,4,2,0,0,0,17,10] >;

C22×C32⋊C9 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes C_9
% in TeX

G:=Group("C2^2xC3^2:C9");
// GroupNames label

G:=SmallGroup(324,82);
// by ID

G=gap.SmallGroup(324,82);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,500,303]);
// Polycyclic

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

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