C2xC2^4:C9 - GroupNames

G = C₂×C₂⁴⋊C₉ order 288 = 2⁵·3²

Direct product of C₂ and C₂⁴⋊C₉

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂×C₂⁴⋊C₉, C₂⁵⋊₂C₉, C₂⁴⋊₄C₁₈, (C₂⁴×C₆).₂C₃, (C₂³×C₆).₆C₆, C₆.₃(C₂²⋊A₄), C₂³⋊₃(C₃.A₄), (C₂²×C₆).₁₂A₄, C₃.(C₂×C₂²⋊A₄), C₂²⋊(C₂×C₃.A₄), (C₂×C₆).₁₃(C₂×A₄), SmallGroup(288,838)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂×C₂⁴⋊C₉

Chief series

C₁ — C₂² — C₂⁴ — C₂³×C₆ — C₂⁴⋊C₉ — C₂×C₂⁴⋊C₉

Lower central

C₂⁴ — C₂×C₂⁴⋊C₉

Upper central

C₁ — C₆

Generators and relations for C₂×C₂⁴⋊C₉
G = < a,b,c,d,e,f | a²=b²=c²=d²=e²=f⁹=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf^-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf^-1=b, fdf^-1=de=ed, fef^-1=d >

Subgroups: 822 in 282 conjugacy classes, 30 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₂², C₂², C₆, C₆, C₂³, C₂³, C₉, C₂×C₆, C₂×C₆, C₂⁴, C₂⁴, C₁₈, C₂²×C₆, C₂²×C₆, C₂⁵, C₃.A₄, C₂³×C₆, C₂³×C₆, C₂×C₃.A₄, C₂⁴×C₆, C₂⁴⋊C₉, C₂×C₂⁴⋊C₉
Quotients: C₁, C₂, C₃, C₆, C₉, A₄, C₁₈, C₂×A₄, C₃.A₄, C₂²⋊A₄, C₂×C₃.A₄, C₂×C₂²⋊A₄, C₂⁴⋊C₉, C₂×C₂⁴⋊C₉

Smallest permutation representation of C₂×C₂⁴⋊C₉
►On 36 points

Generators in S₃₆

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

(1 32)(3 34)(4 35)(6 28)(7 29)(9 31)(10 20)(11 21)(13 23)(14 24)(16 26)(17 27)

(1 32)(2 33)(4 35)(5 36)(7 29)(8 30)(11 21)(12 22)(14 24)(15 25)(17 27)(18 19)

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

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

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

48 conjugacy classes

class	1	2A	2B	···	2K	3A	3B	6A	6B	6C	···	6V	9A	···	9F	18A	···	18F
order	1	2	2	···	2	3	3	6	6	6	···	6	9	···	9	18	···	18
size	1	1	3	···	3	1	1	1	1	3	···	3	16	···	16	16	···	16

48 irreducible representations

dim	1	1	1	1	1	1	3	3	3	3
type	+	+					+	+
image	C₁	C₂	C₃	C₆	C₉	C₁₈	A₄	C₂×A₄	C₃.A₄	C₂×C₃.A₄
kernel	C₂×C₂⁴⋊C₉	C₂⁴⋊C₉	C₂⁴×C₆	C₂³×C₆	C₂⁵	C₂⁴	C₂²×C₆	C₂×C₆	C₂³	C₂²
# reps	1	1	2	2	6	6	5	5	10	10

Matrix representation of C₂×C₂⁴⋊C₉ ►in GL₆(𝔽₁₉)

18	0	0	0	0	0
0	18	0	0	0	0
0	0	18	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

18	0	0	0	0	0
0	18	0	0	0	0
1	0	1	0	0	0
0	0	0	18	0	0
0	0	0	0	18	0
0	0	0	7	0	1

18	0	0	0	0	0
6	1	0	0	0	0
0	0	18	0	0	0
0	0	0	18	0	0
0	0	0	9	1	0
0	0	0	0	0	18

18	0	0	0	0	0
0	18	0	0	0	0
1	0	1	0	0	0
0	0	0	1	0	0
0	0	0	10	18	0
0	0	0	12	0	18

18	0	0	0	0	0
6	1	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	7	0	1

13	17	0	0	0	0
9	6	1	0	0	0
7	1	0	0	0	0
0	0	0	5	18	0
0	0	0	0	14	1
0	0	0	10	13	0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,0,1,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,7,0,0,0,0,18,0,0,0,0,0,0,1],[18,6,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,9,0,0,0,0,0,1,0,0,0,0,0,0,18],[18,0,1,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,10,12,0,0,0,0,18,0,0,0,0,0,0,18],[18,6,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,7,0,0,0,0,18,0,0,0,0,0,0,1],[13,9,7,0,0,0,17,6,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,10,0,0,0,18,14,13,0,0,0,0,1,0] >;

C₂×C₂⁴⋊C₉ in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes C_9

% in TeX

G:=Group("C2xC2^4:C9");

// GroupNames label

G:=SmallGroup(288,838);

// by ID

G=gap.SmallGroup(288,838);

# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,514,956,3036,5305]);

// Polycyclic

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

// generators/relations

G = C2×C24⋊C9 order 288 = 25·32

Direct product of C2 and C24⋊C9

G = C₂×C₂⁴⋊C₉ order 288 = 2⁵·3²

Direct product of C₂ and C₂⁴⋊C₉