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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

Aliases: C22×C27⋊C3, C542C6, C18.4C18, C62.3C9, C9.1C62, C9.(C2×C18), (C2×C54)⋊3C3, C272(C2×C6), C3.3(C6×C18), (C6×C18).9C3, (C2×C18).4C9, C18.6(C3×C6), C6.7(C3×C18), (C3×C6).4C18, C32.(C2×C18), (C3×C18).18C6, (C2×C18).8C32, (C3×C9).4(C2×C6), (C2×C6).12(C3×C9), SmallGroup(324,85)

Series: Derived Chief Lower central Upper central

C1C3 — C22×C27⋊C3
C1C3C9C3×C9C27⋊C3C2×C27⋊C3 — C22×C27⋊C3
C1C3 — C22×C27⋊C3
C1C2×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 [×3], C3, C3, C22, C6 [×3], C6 [×3], C9, C9 [×2], C32, C2×C6, C2×C6, C18 [×9], C3×C6 [×3], C27 [×3], C3×C9, C2×C18, C2×C18 [×2], C62, C54 [×9], C3×C18 [×3], C27⋊C3, C2×C54 [×3], C6×C18, C2×C27⋊C3 [×3], C22×C27⋊C3
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C9 [×3], C32, C2×C6 [×4], C18 [×9], C3×C6 [×3], C3×C9, C2×C18 [×3], C62, C3×C18 [×3], C27⋊C3, C6×C18, C2×C27⋊C3 [×3], C22×C27⋊C3

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

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

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

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

132 conjugacy classes

class 1 2A2B2C3A3B3C3D6A···6F6G···6L9A···9F9G9H9I9J18A···18R18S···18AD27A···27R54A···54BB
order122233336···66···69···9999918···1818···1827···2754···54
size111111331···13···31···133331···13···33···33···3

132 irreducible representations

dim111111111133
type++
imageC1C2C3C3C6C6C9C9C18C18C27⋊C3C2×C27⋊C3
kernelC22×C27⋊C3C2×C27⋊C3C2×C54C6×C18C54C3×C18C2×C18C62C18C3×C6C22C2
# reps13621861263618618

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

108000
010800
001080
000108
,
1000
010800
001080
000108
,
63000
0010
00063
02700
,
45000
0100
00630
00045
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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