C9xC3^2:C6 - GroupNames

G = C₉×C₃²⋊C₆ order 486 = 2·3⁵

Direct product of C₉ and C₃²⋊C₆

direct product, metabelian, supersoluble, monomial

Aliases: C₉×C₃²⋊C₆, He₃⋊₃C₁₈, (C₉×He₃)⋊₁C₂, (C₃²×C₉)⋊₂C₆, C₃²⋊₁(S₃×C₉), (C₃²×C₉)⋊₃S₃, C₃²⋊C₁₈⋊₃C₃, C₃²⋊C₉⋊₁₈C₆, C₃²⋊₁(C₃×C₁₈), (C₃×He₃).₂₂C₆, C₃³.₃₁(C₃×S₃), C₃³.₂₁(C₃×C₆), C₃².₃₁(S₃×C₃²), C₃.₄(S₃×C₃×C₉), (C₉×C₃⋊S₃)⋊₁C₃, C₃⋊S₃⋊₁(C₃×C₉), (C₃×C₉).₁₅(C₃×S₃), (C₃×C₃²⋊C₆).₂C₃, (C₃×C₃⋊S₃).₂C₃², C₃.₁₁(C₃×C₃²⋊C₆), SmallGroup(486,98)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₉×C₃²⋊C₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₉ — C₉×He₃ — C₉×C₃²⋊C₆

Lower central

C₃² — C₉×C₃²⋊C₆

Upper central

C₁ — C₉

Generators and relations for C₉×C₃²⋊C₆
G = < a,b,c,d | a⁹=b³=c³=d⁶=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1c^-1, dcd^-1=c^-1 >

Subgroups: 346 in 100 conjugacy classes, 33 normal (18 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², C₃², C₁₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, C₃×C₉, He₃, He₃, C₃³, C₃³, S₃×C₉, C₃²⋊C₆, C₃×C₁₈, S₃×C₃², C₃×C₃⋊S₃, C₃²⋊C₉, C₃²⋊C₉, C₃²×C₉, C₃²×C₉, C₃×He₃, C₃²⋊C₁₈, S₃×C₃×C₉, C₃×C₃²⋊C₆, C₉×C₃⋊S₃, C₉×He₃, C₉×C₃²⋊C₆
Quotients: C₁, C₂, C₃, S₃, C₆, C₉, C₃², C₁₈, C₃×S₃, C₃×C₆, C₃×C₉, S₃×C₉, C₃²⋊C₆, C₃×C₁₈, S₃×C₃², S₃×C₃×C₉, C₃×C₃²⋊C₆, C₉×C₃²⋊C₆

Smallest permutation representation of C₉×C₃²⋊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 31 44)(2 32 45)(3 33 37)(4 34 38)(5 35 39)(6 36 40)(7 28 41)(8 29 42)(9 30 43)(10 53 23)(11 54 24)(12 46 25)(13 47 26)(14 48 27)(15 49 19)(16 50 20)(17 51 21)(18 52 22)

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

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

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

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

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

90 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3K	3L	···	3T	6A	···	6H	9A	···	9F	9G	···	9L	9M	···	9X	9Y	···	9AP	18A	···	18R
order	1	2	3	3	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	6	···	6	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	C₃²⋊C₆	C₃×C₃²⋊C₆	C₉×C₃²⋊C₆
kernel	C₉×C₃²⋊C₆	C₉×He₃	C₃²⋊C₁₈	C₃×C₃²⋊C₆	C₉×C₃⋊S₃	C₃²⋊C₉	C₃²×C₉	C₃×He₃	C₃²⋊C₆	He₃	C₃²×C₉	C₃×C₉	C₃³	C₃²	C₉	C₃	C₁
# reps	1	1	4	2	2	4	2	2	18	18	1	6	2	18	1	2	6

Matrix representation of C₉×C₃²⋊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

7	0	15	0	0	0
0	0	18	0	0	0
0	11	12	0	0	0
0	0	0	11	13	0
0	0	0	0	8	7
0	0	0	0	18	0

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	0	0	0
0	0	0	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

0	0	0	1	0	13
0	0	0	8	0	8
0	0	0	11	1	18
1	0	13	0	0	0
8	0	8	0	0	0
11	1	18	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],[7,0,0,0,0,0,0,0,11,0,0,0,15,18,12,0,0,0,0,0,0,11,0,0,0,0,0,13,8,18,0,0,0,0,7,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,0,1,8,11,0,0,0,0,0,1,0,0,0,13,8,18,1,8,11,0,0,0,0,0,1,0,0,0,13,8,18,0,0,0] >;

C₉×C₃²⋊C₆ in GAP, Magma, Sage, TeX

C_9\times C_3^2\rtimes C_6

% in TeX

G:=Group("C9xC3^2:C6");

// GroupNames label

G:=SmallGroup(486,98);

// by ID

G=gap.SmallGroup(486,98);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,3244,3250,11669]);

// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^3=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;

// generators/relations

G = C9×C32⋊C6 order 486 = 2·35

Direct product of C9 and C32⋊C6

G = C₉×C₃²⋊C₆ order 486 = 2·3⁵

Direct product of C₉ and C₃²⋊C₆