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## G = C22×D27order 216 = 23·33

### Direct product of C22 and D27

Aliases: C22×D27, C27⋊C23, C54⋊C22, C18.13D6, C6.13D18, (C2×C54)⋊3C2, (C2×C6).4D9, C9.(C22×S3), C3.(C22×D9), (C2×C18).4S3, SmallGroup(216,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C22×D27
 Chief series C1 — C3 — C9 — C27 — D27 — D54 — C22×D27
 Lower central C27 — C22×D27
 Upper central C1 — C22

Generators and relations for C22×D27
G = < a,b,c,d | a2=b2=c27=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 460 in 64 conjugacy classes, 31 normal (9 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, C9, D6, C2×C6, D9, C18, C22×S3, C27, D18, C2×C18, D27, C54, C22×D9, D54, C2×C54, C22×D27
Quotients: C1, C2, C22, S3, C23, D6, D9, C22×S3, D18, D27, C22×D9, D54, C22×D27

Smallest permutation representation of C22×D27
On 108 points
Generators in S108
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 55)(27 56)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)(26 39)(27 38)(55 103)(56 102)(57 101)(58 100)(59 99)(60 98)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 108)(78 107)(79 106)(80 105)(81 104)

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

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

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

C22×D27 is a maximal subgroup of   D54⋊C4
C22×D27 is a maximal quotient of   D1085C2  D42D27  Q83D27

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 9A 9B 9C 18A ··· 18I 27A ··· 27I 54A ··· 54AA order 1 2 2 2 2 2 2 2 3 6 6 6 9 9 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 1 1 27 27 27 27 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D6 D9 D18 D27 D54 kernel C22×D27 D54 C2×C54 C2×C18 C18 C2×C6 C6 C22 C2 # reps 1 6 1 1 3 3 9 9 27

Matrix representation of C22×D27 in GL3(𝔽109) generated by

 108 0 0 0 1 0 0 0 1
,
 108 0 0 0 108 0 0 0 108
,
 1 0 0 0 10 102 0 7 17
,
 1 0 0 0 108 0 0 1 1
G:=sub<GL(3,GF(109))| [108,0,0,0,1,0,0,0,1],[108,0,0,0,108,0,0,0,108],[1,0,0,0,10,7,0,102,17],[1,0,0,0,108,1,0,0,1] >;

C22×D27 in GAP, Magma, Sage, TeX

C_2^2\times D_{27}
% in TeX

G:=Group("C2^2xD27");
// GroupNames label

G:=SmallGroup(216,23);
// by ID

G=gap.SmallGroup(216,23);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,963,381,3604,208,5189]);
// Polycyclic

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

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