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## G = C22×C27⋊C3order 324 = 22·34

### Direct product of C22 and C27⋊C3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C22×C27⋊C3
 Chief series C1 — C3 — C9 — C3×C9 — C27⋊C3 — C2×C27⋊C3 — C22×C27⋊C3
 Lower central C1 — C3 — C22×C27⋊C3
 Upper central C1 — C2×C18 — C22×C27⋊C3

Generators and relations for C22×C27⋊C3
G = < a,b,c,d | a2=b2=c27=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c10 >

Subgroups: 70 in 60 conjugacy classes, 55 normal (12 characteristic)
C1, C2, C3, C3, C22, C6, C6, C9, C9, C32, C2×C6, C2×C6, C18, C3×C6, C27, C3×C9, C2×C18, C2×C18, C62, C54, C3×C18, C27⋊C3, C2×C54, C6×C18, C2×C27⋊C3, C22×C27⋊C3
Quotients: C1, C2, C3, C22, C6, C9, C32, C2×C6, C18, C3×C6, C3×C9, C2×C18, C62, C3×C18, C27⋊C3, C6×C18, C2×C27⋊C3, C22×C27⋊C3

Smallest permutation representation of C22×C27⋊C3
On 108 points
Generators in S108
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 104)(29 105)(30 106)(31 107)(32 108)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(55 102)(56 103)(57 104)(58 105)(59 106)(60 107)(61 108)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 54)(55 73 64)(56 65 74)(58 76 67)(59 68 77)(61 79 70)(62 71 80)(82 91 100)(84 102 93)(85 94 103)(87 105 96)(88 97 106)(90 108 99)

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

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

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

132 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A ··· 6F 6G ··· 6L 9A ··· 9F 9G 9H 9I 9J 18A ··· 18R 18S ··· 18AD 27A ··· 27R 54A ··· 54BB order 1 2 2 2 3 3 3 3 6 ··· 6 6 ··· 6 9 ··· 9 9 9 9 9 18 ··· 18 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 1 1 1 1 3 3 1 ··· 1 3 ··· 3 1 ··· 1 3 3 3 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

132 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C6 C6 C9 C9 C18 C18 C27⋊C3 C2×C27⋊C3 kernel C22×C27⋊C3 C2×C27⋊C3 C2×C54 C6×C18 C54 C3×C18 C2×C18 C62 C18 C3×C6 C22 C2 # reps 1 3 6 2 18 6 12 6 36 18 6 18

Matrix representation of C22×C27⋊C3 in GL4(𝔽109) generated by

 108 0 0 0 0 108 0 0 0 0 108 0 0 0 0 108
,
 1 0 0 0 0 108 0 0 0 0 108 0 0 0 0 108
,
 63 0 0 0 0 0 1 0 0 0 0 63 0 27 0 0
,
 45 0 0 0 0 1 0 0 0 0 63 0 0 0 0 45
G:=sub<GL(4,GF(109))| [108,0,0,0,0,108,0,0,0,0,108,0,0,0,0,108],[1,0,0,0,0,108,0,0,0,0,108,0,0,0,0,108],[63,0,0,0,0,0,0,27,0,1,0,0,0,0,63,0],[45,0,0,0,0,1,0,0,0,0,63,0,0,0,0,45] >;

C22×C27⋊C3 in GAP, Magma, Sage, TeX

C_2^2\times C_{27}\rtimes C_3
% in TeX

G:=Group("C2^2xC27:C3");
// GroupNames label

G:=SmallGroup(324,85);
// by ID

G=gap.SmallGroup(324,85);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,176,735,118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^27=d^3=1,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^10>;
// generators/relations

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