C9xC9:C6 - GroupNames

G = C₉×C₉⋊C₆ order 486 = 2·3⁵

Direct product of C₉ and C₉⋊C₆

direct product, metabelian, supersoluble, monomial

Aliases: C₉×C₉⋊C₆, C₉²⋊₆C₆, 3_-¹⁺²⋊₂C₁₈, C₉⋊C₉⋊₁₄C₆, C₉⋊C₁₈⋊₄C₃, C₉⋊₁(C₃×C₁₈), (C₉×D₉)⋊₁C₃, D₉⋊₁(C₃×C₉), C₃².₄(S₃×C₉), (C₃²×C₉).₅S₃, C₃³.₃₂(C₃×S₃), (C₃×D₉).₂C₃², (C₉×3_-¹⁺²)⋊₁C₂, C₃².₃₃(S₃×C₃²), (C₃×3_-¹⁺²).₁₃C₆, C₃.₅(S₃×C₃×C₉), C₃.₄(C₃×C₉⋊C₆), (C₃×C₉⋊C₆).₃C₃, (C₃×C₉).₁₆(C₃×S₃), (C₃×C₉).₁₁(C₃×C₆), SmallGroup(486,100)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₉ — C₉×C₉⋊C₆

Chief series

C₁ — C₃ — C₉ — C₃×C₉ — C₉² — C₉×3_-¹⁺² — C₉×C₉⋊C₆

Lower central

C₉ — C₉×C₉⋊C₆

Upper central

C₁ — C₉

Generators and relations for C₉×C₉⋊C₆
G = < a,b,c | a⁹=b⁹=c⁶=1, ab=ba, ac=ca, cbc^-1=b² >

Subgroups: 238 in 82 conjugacy classes, 33 normal (18 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², C₃², D₉, C₁₈, C₃×S₃, C₃×C₆, C₃×C₉, C₃×C₉, C₃×C₉, 3_-¹⁺², 3_-¹⁺², C₃³, C₃×D₉, S₃×C₉, C₉⋊C₆, C₃×C₁₈, S₃×C₃², C₉², C₉², C₃²⋊C₉, C₉⋊C₉, C₉⋊C₉, C₃²×C₉, C₃×3_-¹⁺², C₉×D₉, C₉⋊C₁₈, S₃×C₃×C₉, C₃×C₉⋊C₆, C₉×3_-¹⁺², C₉×C₉⋊C₆
Quotients: C₁, C₂, C₃, S₃, C₆, C₉, C₃², C₁₈, C₃×S₃, C₃×C₆, C₃×C₉, S₃×C₉, C₉⋊C₆, C₃×C₁₈, S₃×C₃², S₃×C₃×C₉, C₃×C₉⋊C₆, C₉×C₉⋊C₆

Smallest permutation representation of C₉×C₉⋊C₆
►On 54 points

Generators in S₅₄

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

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

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

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

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

90 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3K	6A	···	6H	9A	···	9F	9G	···	9L	9M	···	9X	9Y	···	9AY	18A	···	18R
order	1	2	3	3	3	3	3	3	···	3	6	···	6	9	···	9	9	···	9	9	···	9	9	···	9	18	···	18
size	1	9	1	1	2	2	2	3	···	3	9	···	9	1	···	1	2	···	2	3	···	3	6	···	6	9	···	9

90 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	2	2	6	6	6
type	+	+									+				+
image	C₁	C₂	C₃	C₃	C₃	C₆	C₆	C₆	C₉	C₁₈	S₃	C₃×S₃	C₃×S₃	S₃×C₉	C₉⋊C₆	C₃×C₉⋊C₆	C₉×C₉⋊C₆
kernel	C₉×C₉⋊C₆	C₉×3_-¹⁺²	C₉×D₉	C₉⋊C₁₈	C₃×C₉⋊C₆	C₉²	C₉⋊C₉	C₃×3_-¹⁺²	C₉⋊C₆	3_-¹⁺²	C₃²×C₉	C₃×C₉	C₃³	C₃²	C₉	C₃	C₁
# reps	1	1	2	4	2	2	4	2	18	18	1	6	2	18	1	2	6

Matrix representation of C₉×C₉⋊C₆ ►in GL₆(𝔽₁₉)

6	0	0	0	0	0
0	6	0	0	0	0
0	0	6	0	0	0
0	0	0	6	0	0
0	0	0	0	6	0
0	0	0	0	0	6

18	11	8	3	14	16
18	0	8	14	17	0
6	0	1	14	17	7
0	0	0	0	0	1
0	0	0	7	0	0
0	0	0	0	7	0

16	2	6	14	9	8
11	14	10	3	5	3
0	0	0	0	0	11
4	0	7	3	5	11
0	6	1	12	5	2
0	0	11	0	0	0

G:=sub<GL(6,GF(19))| [6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[18,18,6,0,0,0,11,0,0,0,0,0,8,8,1,0,0,0,3,14,14,0,7,0,14,17,17,0,0,7,16,0,7,1,0,0],[16,11,0,4,0,0,2,14,0,0,6,0,6,10,0,7,1,11,14,3,0,3,12,0,9,5,0,5,5,0,8,3,11,11,2,0] >;

C₉×C₉⋊C₆ in GAP, Magma, Sage, TeX

C_9\times C_9\rtimes C_6

% in TeX

G:=Group("C9xC9:C6");

// GroupNames label

G:=SmallGroup(486,100);

// by ID

G=gap.SmallGroup(486,100);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,8104,3250,208,11669]);

// Polycyclic

G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;

// generators/relations

G = C9×C9⋊C6 order 486 = 2·35

Direct product of C9 and C9⋊C6

G = C₉×C₉⋊C₆ order 486 = 2·3⁵

Direct product of C₉ and C₉⋊C₆