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G = C4×D27order 216 = 23·33

Direct product of C4 and D27

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D27, D54.C2, C1082C2, C36.5S3, C12.5D9, C18.9D6, C6.9D18, C2.1D54, Dic272C2, C54.2C22, C9.(C4×S3), C3.(C4×D9), C271(C2×C4), SmallGroup(216,5)

Series: Derived Chief Lower central Upper central

C1C27 — C4×D27
C1C3C9C27C54D54 — C4×D27
C27 — C4×D27
C1C4

Generators and relations for C4×D27
 G = < a,b,c | a4=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

27C2
27C2
27C22
27C4
9S3
9S3
27C2×C4
9D6
9Dic3
3D9
3D9
9C4×S3
3D18
3Dic9
3C4×D9

Smallest permutation representation of C4×D27
On 108 points
Generators in S108
(1 83 48 55)(2 84 49 56)(3 85 50 57)(4 86 51 58)(5 87 52 59)(6 88 53 60)(7 89 54 61)(8 90 28 62)(9 91 29 63)(10 92 30 64)(11 93 31 65)(12 94 32 66)(13 95 33 67)(14 96 34 68)(15 97 35 69)(16 98 36 70)(17 99 37 71)(18 100 38 72)(19 101 39 73)(20 102 40 74)(21 103 41 75)(22 104 42 76)(23 105 43 77)(24 106 44 78)(25 107 45 79)(26 108 46 80)(27 82 47 81)
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(55 83)(56 82)(57 108)(58 107)(59 106)(60 105)(61 104)(62 103)(63 102)(64 101)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)

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

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

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

C4×D27 is a maximal subgroup of   C8⋊D27  D1085C2  D42D27  Q83D27
C4×D27 is a maximal quotient of   C8⋊D27  Dic27⋊C4  D54⋊C4

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 9A9B9C12A12B18A18B18C27A···27I36A···36F54A···54I108A···108R
order1222344446999121218181827···2736···3654···54108···108
size11272721127272222222222···22···22···22···2

60 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4S3D6D9C4×S3D18D27C4×D9D54C4×D27
kernelC4×D27Dic27C108D54D27C36C18C12C9C6C4C3C2C1
# reps111141132396918

Matrix representation of C4×D27 in GL2(𝔽109) generated by

760
076
,
8087
2258
,
2732
5982
G:=sub<GL(2,GF(109))| [76,0,0,76],[80,22,87,58],[27,59,32,82] >;

C4×D27 in GAP, Magma, Sage, TeX

C_4\times D_{27}
% in TeX

G:=Group("C4xD27");
// GroupNames label

G:=SmallGroup(216,5);
// by ID

G=gap.SmallGroup(216,5);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,963,381,3604,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^4=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D27 in TeX

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