C3xDic27 - GroupNames

G = C₃xDic₂₇ order 324 = 2²·3⁴

Direct product of C₃ and Dic₂₇

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

Aliases: C₃xDic₂₇, C₂₇:₃C₁₂, C₅₄.₃C₆, C₆.₄D₂₇, C₃².₂Dic₉, (C₃xC₂₇):₂C₄, C₂.(C₃xD₂₇), C₆.₂(C₃xD₉), (C₃xC₆).₅D₉, C₁₈.₃(C₃xS₃), (C₃xC₅₄).₂C₂, (C₃xC₁₈).₁₈S₃, C₃.₂(C₃xDic₉), (C₃xC₉).₅Dic₃, C₉.₁(C₃xDic₃), SmallGroup(324,10)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₇ — C₃xDic₂₇

Chief series

C₁ — C₃ — C₉ — C₂₇ — C₅₄ — C₃xC₅₄ — C₃xDic₂₇

Lower central

C₂₇ — C₃xDic₂₇

Upper central

C₁ — C₆

Generators and relations for C₃xDic₂₇
G = < a,b,c | a³=b⁵⁴=1, c²=b²⁷, ab=ba, ac=ca, cbc^-1=b^-1 >

Subgroups: 108 in 30 conjugacy classes, 18 normal (all characteristic)
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, Dic₃, C₁₂, D₉, C₃xS₃, Dic₉, C₃xDic₃, D₂₇, C₃xD₉, Dic₂₇, C₃xDic₉, C₃xD₂₇, C₃xDic₂₇

C₁

C₂

C₃

₂C₃

₂₇C₄

C₆

₂C₆

C₉

C₃²

₂C₉

₉Dic₃

₂₇C₁₂

C₁₈

C₃xC₆

₂C₁₈

C₂₇

C₃xC₉

₂C₂₇

₃Dic₉

₉C₃xDic₃

C₅₄

C₃xC₁₈

₂C₅₄

C₃xC₂₇

Dic₂₇

₃C₃xDic₉

C₃xC₅₄

C₃xDic₂₇

Smallest permutation representation of C₃xDic₂₇
►On 108 points

Generators in S₁₀₈

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

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

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

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

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

90 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	6D	6E	9A	···	9I	12A	12B	12C	12D	18A	···	18I	27A	···	27AA	54A	···	54AA
order	1	2	3	3	3	3	3	4	4	6	6	6	6	6	9	···	9	12	12	12	12	18	···	18	27	···	27	54	···	54
size	1	1	1	1	2	2	2	27	27	1	1	2	2	2	2	···	2	27	27	27	27	2	···	2	2	···	2	2	···	2

90 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+					+	-		+		-	+		-
image	C₁	C₂	C₃	C₄	C₆	C₁₂	S₃	Dic₃	C₃xS₃	D₉	C₃xDic₃	Dic₉	D₂₇	C₃xD₉	Dic₂₇	C₃xDic₉	C₃xD₂₇	C₃xDic₂₇
kernel	C₃xDic₂₇	C₃xC₅₄	Dic₂₇	C₃xC₂₇	C₅₄	C₂₇	C₃xC₁₈	C₃xC₉	C₁₈	C₃xC₆	C₉	C₃²	C₆	C₆	C₃	C₃	C₂	C₁
# reps	1	1	2	2	2	4	1	1	2	3	2	3	9	6	9	6	18	18

Matrix representation of C₃xDic₂₇ ►in GL₃(F₁₀₉) generated by

45	0	0
0	45	0
0	0	45

108	0	0
0	26	0
0	106	21

76	0	0
0	59	62
0	81	50

G:=sub<GL(3,GF(109))| [45,0,0,0,45,0,0,0,45],[108,0,0,0,26,106,0,0,21],[76,0,0,0,59,81,0,62,50] >;

C₃xDic₂₇ in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{27}

% in TeX

G:=Group("C3xDic27");

// GroupNames label

G:=SmallGroup(324,10);

// by ID

G=gap.SmallGroup(324,10);

# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,1443,381,5404,208,7781]);

// Polycyclic

G:=Group<a,b,c|a^3=b^54=1,c^2=b^27,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

Export

Subgroup lattice of C₃xDic₂₇ in TeX

G = C3xDic27 order 324 = 22·34

Direct product of C3 and Dic27

G = C₃xDic₂₇ order 324 = 2²·3⁴

Direct product of C₃ and Dic₂₇