C3xC27:C6 - GroupNames

G = C₃×C₂₇⋊C₆ order 486 = 2·3⁵

Direct product of C₃ and C₂₇⋊C₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₂₇⋊C₆, D₂₇⋊C₃², C₃³.₄D₉, C₂₇⋊(C₃×C₆), (C₃×D₂₇)⋊C₃, C₂₇⋊C₃⋊₄C₆, (C₃×C₂₇)⋊₄C₆, C₉.₄(S₃×C₃²), C₃².₈(C₃×D₉), C₃.₃(C₃²×D₉), (C₃²×C₉).₁₈S₃, (C₃×C₂₇⋊C₃)⋊₁C₂, (C₃×C₉).₂₂(C₃×S₃), SmallGroup(486,113)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₇ — C₃×C₂₇⋊C₆

Chief series

C₁ — C₃ — C₉ — C₂₇ — C₃×C₂₇ — C₃×C₂₇⋊C₃ — C₃×C₂₇⋊C₆

Lower central

C₂₇ — C₃×C₂₇⋊C₆

Upper central

C₁ — C₃

Generators and relations for C₃×C₂₇⋊C₆
G = < a,b,c | a³=b²⁷=c⁶=1, ab=ba, ac=ca, cbc^-1=b¹⁷ >

Subgroups: 312 in 66 conjugacy classes, 26 normal (14 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², C₃², D₉, C₃×S₃, C₃×C₆, C₂₇, C₂₇, C₃×C₉, C₃×C₉, C₃×C₉, C₃³, D₂₇, C₃×D₉, S₃×C₃², C₃×C₂₇, C₃×C₂₇, C₂₇⋊C₃, C₂₇⋊C₃, C₃²×C₉, C₃×D₂₇, C₂₇⋊C₆, C₃²×D₉, C₃×C₂₇⋊C₃, C₃×C₂₇⋊C₆
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², D₉, C₃×S₃, C₃×C₆, C₃×D₉, S₃×C₃², C₂₇⋊C₆, C₃²×D₉, C₃×C₂₇⋊C₆

Smallest permutation representation of C₃×C₂₇⋊C₆
►On 54 points

Generators in S₅₄

(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)

(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)

(1 30 19 39 10 48)(2 38)(3 46 12 37 21 28)(4 54 22 36 13 45)(5 35)(6 43 15 34 24 52)(7 51 25 33 16 42)(8 32)(9 40 18 31 27 49)(11 29)(14 53)(17 50)(20 47)(23 44)(26 41)

G:=sub<Sym(54)| (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (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), (1,30,19,39,10,48)(2,38)(3,46,12,37,21,28)(4,54,22,36,13,45)(5,35)(6,43,15,34,24,52)(7,51,25,33,16,42)(8,32)(9,40,18,31,27,49)(11,29)(14,53)(17,50)(20,47)(23,44)(26,41)>;

G:=Group( (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (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), (1,30,19,39,10,48)(2,38)(3,46,12,37,21,28)(4,54,22,36,13,45)(5,35)(6,43,15,34,24,52)(7,51,25,33,16,42)(8,32)(9,40,18,31,27,49)(11,29)(14,53)(17,50)(20,47)(23,44)(26,41) );

G=PermutationGroup([[(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18),(28,37,46),(29,38,47),(30,39,48),(31,40,49),(32,41,50),(33,42,51),(34,43,52),(35,44,53),(36,45,54)], [(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)], [(1,30,19,39,10,48),(2,38),(3,46,12,37,21,28),(4,54,22,36,13,45),(5,35),(6,43,15,34,24,52),(7,51,25,33,16,42),(8,32),(9,40,18,31,27,49),(11,29),(14,53),(17,50),(20,47),(23,44),(26,41)]])

63 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3K	6A	···	6H	9A	···	9I	9J	···	9O	27A	···	27AA
order	1	2	3	3	3	3	3	3	···	3	6	···	6	9	···	9	9	···	9	27	···	27
size	1	27	1	1	2	2	2	3	···	3	27	···	27	2	···	2	6	···	6	6	···	6

63 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	6	6
type	+	+					+		+		+
image	C₁	C₂	C₃	C₃	C₆	C₆	S₃	C₃×S₃	D₉	C₃×D₉	C₂₇⋊C₆	C₃×C₂₇⋊C₆
kernel	C₃×C₂₇⋊C₆	C₃×C₂₇⋊C₃	C₃×D₂₇	C₂₇⋊C₆	C₃×C₂₇	C₂₇⋊C₃	C₃²×C₉	C₃×C₉	C₃³	C₃²	C₃	C₁
# reps	1	1	2	6	2	6	1	8	3	24	3	6

Matrix representation of C₃×C₂₇⋊C₆ ►in GL₆(𝔽₁₀₉)

63	0	0	0	0	0
0	63	0	0	0	0
0	0	63	0	0	0
0	0	0	63	0	0
0	0	0	0	63	0
0	0	0	0	0	63

0	63	0	0	0	0
0	0	63	0	0	0
38	0	0	0	0	0
0	0	0	0	0	66
0	0	0	45	0	0
0	0	0	0	45	0

0	0	0	45	0	0
0	0	0	0	63	0
0	0	0	0	0	1
45	0	0	0	0	0
0	63	0	0	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(109))| [63,0,0,0,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,63],[0,0,38,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,45,0,0,0,66,0,0],[0,0,0,45,0,0,0,0,0,0,63,0,0,0,0,0,0,1,45,0,0,0,0,0,0,63,0,0,0,0,0,0,1,0,0,0] >;

C₃×C₂₇⋊C₆ in GAP, Magma, Sage, TeX

C_3\times C_{27}\rtimes C_6

% in TeX

G:=Group("C3xC27:C6");

// GroupNames label

G:=SmallGroup(486,113);

// by ID

G=gap.SmallGroup(486,113);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,2163,2169,381,8104,208,11669]);

// Polycyclic

G:=Group<a,b,c|a^3=b^27=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^17>;

// generators/relations

G = C3×C27⋊C6 order 486 = 2·35

Direct product of C3 and C27⋊C6

G = C₃×C₂₇⋊C₆ order 486 = 2·3⁵

Direct product of C₃ and C₂₇⋊C₆