C2xC3^2:C18 - GroupNames

G = C₂×C₃²⋊C₁₈ order 324 = 2²·3⁴

Direct product of C₂ and C₃²⋊C₁₈

direct product, metabelian, supersoluble, monomial

Aliases: C₂×C₃²⋊C₁₈, C₃⋊S₃⋊C₁₈, (C₃×C₆)⋊C₁₈, (C₃×C₉)⋊₅D₆, (C₃×C₁₈)⋊₁S₃, C₆.₅(S₃×C₉), C₃²⋊(C₂×C₁₈), C₃.₂(S₃×C₁₈), C₃²⋊C₉⋊₂C₂², (C₃²×C₆).₃C₆, C₃³.₁(C₂×C₆), C₃².₁₄(S₃×C₆), C₆.₁₄(C₃²⋊C₆), (C₂×C₃⋊S₃)⋊C₉, (C₃×C₃⋊S₃).C₆, (C₆×C₃⋊S₃).C₃, (C₂×C₃²⋊C₉)⋊₁C₂, (C₃×C₆).₃₀(C₃×S₃), C₃.₅(C₂×C₃²⋊C₆), SmallGroup(324,62)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×C₃²⋊C₁₈

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²⋊C₉ — C₃²⋊C₁₈ — C₂×C₃²⋊C₁₈

Lower central

C₃² — C₂×C₃²⋊C₁₈

Upper central

C₁ — C₆

Generators and relations for C₂×C₃²⋊C₁₈
G = < a,b,c,d | a²=b³=c³=d¹⁸=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1c^-1, dcd^-1=c^-1 >

Subgroups: 253 in 67 conjugacy classes, 25 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², S₃, C₆, C₆, C₉, C₃², C₃², D₆, C₂×C₆, C₁₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₉, C₃×C₉, C₃³, C₂×C₁₈, S₃×C₆, C₂×C₃⋊S₃, S₃×C₉, C₃×C₁₈, C₃×C₁₈, C₃×C₃⋊S₃, C₃²×C₆, C₃²⋊C₉, S₃×C₁₈, C₆×C₃⋊S₃, C₃²⋊C₁₈, C₂×C₃²⋊C₉, C₂×C₃²⋊C₁₈
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₉, D₆, C₂×C₆, C₁₈, C₃×S₃, C₂×C₁₈, S₃×C₆, S₃×C₉, C₃²⋊C₆, S₃×C₁₈, C₂×C₃²⋊C₆, C₃²⋊C₁₈, C₂×C₃²⋊C₁₈

Smallest permutation representation of C₂×C₃²⋊C₁₈
►On 36 points

Generators in S₃₆

(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)

(2 8 14)(3 9 15)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(22 28 34)(23 29 35)

(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)(19 31 25)(20 26 32)(21 33 27)(22 28 34)(23 35 29)(24 30 36)

(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)

G:=sub<Sym(36)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (2,8,14)(3,9,15)(5,17,11)(6,18,12)(19,31,25)(20,32,26)(22,28,34)(23,29,35), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18)(19,31,25)(20,26,32)(21,33,27)(22,28,34)(23,35,29)(24,30,36), (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)>;

G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (2,8,14)(3,9,15)(5,17,11)(6,18,12)(19,31,25)(20,32,26)(22,28,34)(23,29,35), (1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,17,11)(6,12,18)(19,31,25)(20,26,32)(21,33,27)(22,28,34)(23,35,29)(24,30,36), (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) );

G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26)], [(2,8,14),(3,9,15),(5,17,11),(6,18,12),(19,31,25),(20,32,26),(22,28,34),(23,29,35)], [(1,13,7),(2,8,14),(3,15,9),(4,10,16),(5,17,11),(6,12,18),(19,31,25),(20,26,32),(21,33,27),(22,28,34),(23,35,29),(24,30,36)], [(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)]])

60 conjugacy classes

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

60 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	6	6	6	6
type	+	+	+							+	+					+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	C₉	C₁₈	C₁₈	S₃	D₆	C₃×S₃	S₃×C₆	S₃×C₉	S₃×C₁₈	C₃²⋊C₆	C₂×C₃²⋊C₆	C₃²⋊C₁₈	C₂×C₃²⋊C₁₈
kernel	C₂×C₃²⋊C₁₈	C₃²⋊C₁₈	C₂×C₃²⋊C₉	C₆×C₃⋊S₃	C₃×C₃⋊S₃	C₃²×C₆	C₂×C₃⋊S₃	C₃⋊S₃	C₃×C₆	C₃×C₁₈	C₃×C₉	C₃×C₆	C₃²	C₆	C₃	C₆	C₃	C₂	C₁
# reps	1	2	1	2	4	2	6	12	6	1	1	2	2	6	6	1	1	2	2

Matrix representation of C₂×C₃²⋊C₁₈ ►in GL₆(𝔽₁₉)

18	0	0	0	0	0
0	18	0	0	0	0
0	0	18	0	0	0
0	0	0	18	0	0
0	0	0	0	18	0
0	0	0	0	0	18

1	18	7	0	3	9
0	7	0	0	0	0
0	0	11	0	0	0
0	0	0	1	0	0
0	0	0	0	11	0
0	0	0	0	0	7

11	0	0	11	14	13
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

4	12	16	16	0	11
0	0	0	0	0	12
0	0	0	18	0	0
0	12	0	0	0	0
0	0	12	0	0	0
9	11	7	1	3	15

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,18,7,0,0,0,0,7,0,11,0,0,0,0,0,0,1,0,0,3,0,0,0,11,0,9,0,0,0,0,7],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,11,0,0,7,0,0,14,0,0,0,7,0,13,0,0,0,0,7],[4,0,0,0,0,9,12,0,0,12,0,11,16,0,0,0,12,7,16,0,18,0,0,1,0,0,0,0,0,3,11,12,0,0,0,15] >;

C₂×C₃²⋊C₁₈ in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes C_{18}

% in TeX

G:=Group("C2xC3^2:C18");

// GroupNames label

G:=SmallGroup(324,62);

// by ID

G=gap.SmallGroup(324,62);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,2164,1096,7781]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^3=d^18=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 = C2×C32⋊C18 order 324 = 22·34

Direct product of C2 and C32⋊C18

G = C₂×C₃²⋊C₁₈ order 324 = 2²·3⁴

Direct product of C₂ and C₃²⋊C₁₈