Dic3xC27 - GroupNames

G = Dic₃xC₂₇ order 324 = 2²·3⁴

Direct product of C₂₇ and Dic₃

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

Aliases: Dic₃xC₂₇, C₃:C₁₀₈, C₆.C₅₄, C₅₄.₄S₃, C₃².₂C₃₆, (C₃xC₂₇):₁C₄, C₂.(S₃xC₂₇), C₆.₈(S₃xC₉), (C₃xC₆).₅C₁₈, (C₃xC₅₄).₁C₂, (C₃xC₉).₅C₁₂, (C₃xDic₃).C₉, (C₉xDic₃).C₃, C₁₈.₁₀(C₃xS₃), (C₃xC₁₈).₂₁C₆, C₃.₄(C₉xDic₃), C₉.₄(C₃xDic₃), SmallGroup(324,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — Dic₃xC₂₇

Chief series

C₁ — C₃ — C₃² — C₃xC₉ — C₃xC₁₈ — C₃xC₅₄ — Dic₃xC₂₇

Lower central

C₃ — Dic₃xC₂₇

Upper central

C₁ — C₅₄

Generators and relations for Dic₃xC₂₇
G = < a,b,c | a²⁷=b⁶=1, c²=b³, ab=ba, ac=ca, cbc^-1=b^-1 >

Subgroups: 44 in 30 conjugacy classes, 20 normal (all characteristic)
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, C₉, Dic₃, C₁₂, C₁₈, C₃xS₃, C₂₇, C₃₆, C₃xDic₃, C₅₄, S₃xC₉, C₁₀₈, C₉xDic₃, S₃xC₂₇, Dic₃xC₂₇

C₁

C₂

C₃

₂C₃

₃C₄

C₆

₂C₆

C₃²

C₉

₂C₉

Dic₃

₃C₁₂

C₃xC₆

C₁₈

₂C₁₈

C₃xC₉

C₂₇

₂C₂₇

C₃xDic₃

₃C₃₆

C₃xC₁₈

C₅₄

₂C₅₄

C₃xC₂₇

C₉xDic₃

₃C₁₀₈

C₃xC₅₄

Dic₃xC₂₇

Smallest permutation representation of Dic₃xC₂₇
►On 108 points

Generators in S₁₀₈

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

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

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

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

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

162 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	6D	6E	9A	···	9F	9G	···	9L	12A	12B	12C	12D	18A	···	18F	18G	···	18L	27A	···	27R	27S	···	27AJ	36A	···	36L	54A	···	54R	54S	···	54AJ	108A	···	108AJ
order	1	2	3	3	3	3	3	4	4	6	6	6	6	6	9	···	9	9	···	9	12	12	12	12	18	···	18	18	···	18	27	···	27	27	···	27	36	···	36	54	···	54	54	···	54	108	···	108
size	1	1	1	1	2	2	2	3	3	1	1	2	2	2	1	···	1	2	···	2	3	3	3	3	1	···	1	2	···	2	1	···	1	2	···	2	3	···	3	1	···	1	2	···	2	3	···	3

162 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2
type	+	+											+	-
image	C₁	C₂	C₃	C₄	C₆	C₉	C₁₂	C₁₈	C₂₇	C₃₆	C₅₄	C₁₀₈	S₃	Dic₃	C₃xS₃	C₃xDic₃	S₃xC₉	C₉xDic₃	S₃xC₂₇	Dic₃xC₂₇
kernel	Dic₃xC₂₇	C₃xC₅₄	C₉xDic₃	C₃xC₂₇	C₃xC₁₈	C₃xDic₃	C₃xC₉	C₃xC₆	Dic₃	C₃²	C₆	C₃	C₅₄	C₂₇	C₁₈	C₉	C₆	C₃	C₂	C₁
# reps	1	1	2	2	2	6	4	6	18	12	18	36	1	1	2	2	6	6	18	18

Matrix representation of Dic₃xC₂₇ ►in GL₃(F₁₀₉) generated by

78	0	0
0	7	0
0	0	7

108	0	0
0	45	0
0	105	63

33	0	0
0	75	44
0	1	34

G:=sub<GL(3,GF(109))| [78,0,0,0,7,0,0,0,7],[108,0,0,0,45,105,0,0,63],[33,0,0,0,75,1,0,44,34] >;

Dic₃xC₂₇ in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{27}

% in TeX

G:=Group("Dic3xC27");

// GroupNames label

G:=SmallGroup(324,11);

// by ID

G=gap.SmallGroup(324,11);

# by ID

G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,93,7781]);

// Polycyclic

G:=Group<a,b,c|a^27=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

Export

Subgroup lattice of Dic₃xC₂₇ in TeX

G = Dic3xC27 order 324 = 22·34

Direct product of C27 and Dic3

G = Dic₃xC₂₇ order 324 = 2²·3⁴

Direct product of C₂₇ and Dic₃