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## G = C32×Dic9order 324 = 22·34

### Direct product of C32 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C32×Dic9
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C32×C18 — C32×Dic9
 Lower central C9 — C32×Dic9
 Upper central C1 — C3×C6

Generators and relations for C32×Dic9
G = < a,b,c,d | a3=b3=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 178 in 86 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C9, C32, C32, C32, Dic3, C12, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C33, Dic9, C3×Dic3, C3×C12, C3×C18, C3×C18, C32×C6, C32×C9, C3×Dic9, C32×Dic3, C32×C18, C32×Dic9
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, D9, C3×S3, C3×C6, Dic9, C3×Dic3, C3×C12, C3×D9, S3×C32, C3×Dic9, C32×Dic3, C32×D9, C32×Dic9

Smallest permutation representation of C32×Dic9
On 108 points
Generators in S108
(1 45 21)(2 46 22)(3 47 23)(4 48 24)(5 49 25)(6 50 26)(7 51 27)(8 52 28)(9 53 29)(10 54 30)(11 37 31)(12 38 32)(13 39 33)(14 40 34)(15 41 35)(16 42 36)(17 43 19)(18 44 20)(55 101 89)(56 102 90)(57 103 73)(58 104 74)(59 105 75)(60 106 76)(61 107 77)(62 108 78)(63 91 79)(64 92 80)(65 93 81)(66 94 82)(67 95 83)(68 96 84)(69 97 85)(70 98 86)(71 99 87)(72 100 88)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 79 85)(74 80 86)(75 81 87)(76 82 88)(77 83 89)(78 84 90)(91 97 103)(92 98 104)(93 99 105)(94 100 106)(95 101 107)(96 102 108)
(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 72 10 63)(2 71 11 62)(3 70 12 61)(4 69 13 60)(5 68 14 59)(6 67 15 58)(7 66 16 57)(8 65 17 56)(9 64 18 55)(19 90 28 81)(20 89 29 80)(21 88 30 79)(22 87 31 78)(23 86 32 77)(24 85 33 76)(25 84 34 75)(26 83 35 74)(27 82 36 73)(37 108 46 99)(38 107 47 98)(39 106 48 97)(40 105 49 96)(41 104 50 95)(42 103 51 94)(43 102 52 93)(44 101 53 92)(45 100 54 91)

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

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

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

108 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 4A 4B 6A ··· 6H 6I ··· 6Q 9A ··· 9AA 12A ··· 12P 18A ··· 18AA order 1 2 3 ··· 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 1 ··· 1 2 ··· 2 9 9 1 ··· 1 2 ··· 2 2 ··· 2 9 ··· 9 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 D9 C3×S3 Dic9 C3×Dic3 C3×D9 C3×Dic9 kernel C32×Dic9 C32×C18 C3×Dic9 C32×C9 C3×C18 C3×C9 C32×C6 C33 C3×C6 C3×C6 C32 C32 C6 C3 # reps 1 1 8 2 8 16 1 1 3 8 3 8 24 24

Matrix representation of C32×Dic9 in GL5(𝔽37)

 1 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 1
,
 26 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 26 0 0 0 0 0 26
,
 36 0 0 0 0 0 0 1 0 0 0 36 36 0 0 0 0 0 12 0 0 0 0 20 34
,
 31 0 0 0 0 0 1 1 0 0 0 0 36 0 0 0 0 0 25 9 0 0 0 17 12

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[26,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,26],[36,0,0,0,0,0,0,36,0,0,0,1,36,0,0,0,0,0,12,20,0,0,0,0,34],[31,0,0,0,0,0,1,0,0,0,0,1,36,0,0,0,0,0,25,17,0,0,0,9,12] >;

C32×Dic9 in GAP, Magma, Sage, TeX

C_3^2\times {\rm Dic}_9
% in TeX

G:=Group("C3^2xDic9");
// GroupNames label

G:=SmallGroup(324,90);
// by ID

G=gap.SmallGroup(324,90);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,5404,208,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^18=1,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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