C2xC9.4He3 - GroupNames

G = C₂×C₉.₄He₃ order 486 = 2·3⁵

Direct product of C₂ and C₉.₄He₃

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Aliases: C₂×C₉.₄He₃, C₁₈.₃He₃, C₃³.₄C₁₈, C₁₈.₄3_-¹⁺², C₂₇⋊C₃⋊₁C₆, (C₃×C₁₈).₃C₉, (C₃×C₉).₅C₁₈, C₉.₃(C₂×He₃), (C₃²×C₆).₂C₉, C₆.₆(C₃²⋊C₉), (C₃²×C₉).₂₂C₆, (C₃²×C₁₈).₁₀C₃, C₃².₁₀(C₃×C₁₈), (C₃×C₁₈).₂₃C₃², C₉.₄(C₂×3_-¹⁺²), (C₂×C₂₇⋊C₃)⋊₁C₃, (C₃×C₆).₁₀(C₃×C₉), (C₃×C₉).₃₂(C₃×C₆), C₃.₆(C₂×C₃²⋊C₉), SmallGroup(486,76)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×C₉.₄He₃

Chief series

C₁ — C₃ — C₉ — C₃×C₉ — C₃²×C₉ — C₉.₄He₃ — C₂×C₉.₄He₃

Lower central

C₁ — C₃ — C₃² — C₂×C₉.₄He₃

Upper central

C₁ — C₁₈ — C₃×C₁₈ — C₂×C₉.₄He₃

Generators and relations for C₂×C₉.₄He₃
G = < a,b,c,d,e | a²=b⁹=c³=d³=1, e³=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=b³cd^-1, ede^-1=b⁶d >

Subgroups: 126 in 62 conjugacy classes, 30 normal (20 characteristic)
C₁, C₂, C₃, C₃, C₆, C₆, C₉, C₉, C₉, C₃², C₃², C₁₈, C₁₈, C₁₈, C₃×C₆, C₃×C₆, C₂₇, C₃×C₉, C₃×C₉, C₃³, C₅₄, C₃×C₁₈, C₃×C₁₈, C₃²×C₆, C₂₇⋊C₃, C₃²×C₉, C₂×C₂₇⋊C₃, C₃²×C₁₈, C₉.₄He₃, C₂×C₉.₄He₃
Quotients: C₁, C₂, C₃, C₆, C₉, C₃², C₁₈, C₃×C₆, C₃×C₉, He₃, 3_-¹⁺², C₃×C₁₈, C₂×He₃, C₂×3_-¹⁺², C₃²⋊C₉, C₂×C₃²⋊C₉, C₉.₄He₃, C₂×C₉.₄He₃

Smallest permutation representation of C₂×C₉.₄He₃
►On 54 points

Generators in S₅₄

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

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

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

(1 19 10)(2 11 20)(4 22 13)(5 14 23)(7 25 16)(8 17 26)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(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)

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

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

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

102 conjugacy classes

class	1	2	3A	3B	3C	···	3J	6A	6B	6C	···	6J	9A	···	9F	9G	···	9V	18A	···	18F	18G	···	18V	27A	···	27R	54A	···	54R
order	1	2	3	3	3	···	3	6	6	6	···	6	9	···	9	9	···	9	18	···	18	18	···	18	27	···	27	54	···	54
size	1	1	1	1	3	···	3	1	1	3	···	3	1	···	1	3	···	3	1	···	1	3	···	3	9	···	9	9	···	9

102 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	3	3	3	3	3	3
type	+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	C₉	C₉	C₁₈	C₁₈	He₃	3_-¹⁺²	C₂×He₃	C₂×3_-¹⁺²	C₉.₄He₃	C₂×C₉.₄He₃
kernel	C₂×C₉.₄He₃	C₉.₄He₃	C₂×C₂₇⋊C₃	C₃²×C₁₈	C₂₇⋊C₃	C₃²×C₉	C₃×C₁₈	C₃²×C₆	C₃×C₉	C₃³	C₁₈	C₁₈	C₉	C₉	C₂	C₁
# reps	1	1	6	2	6	2	12	6	12	6	2	4	2	4	18	18

Matrix representation of C₂×C₉.₄He₃ ►in GL₄(𝔽₁₀₉) generated by

108	0	0	0
0	1	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	105	0	0
0	0	105	0
0	0	0	105

63	0	0	0
0	63	45	26
0	0	45	0
0	0	0	45

1	0	0	0
0	63	46	26
0	0	1	0
0	0	0	45

1	0	0	0
0	7	14	50
0	0	0	1
0	79	34	102

G:=sub<GL(4,GF(109))| [108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,105,0,0,0,0,105,0,0,0,0,105],[63,0,0,0,0,63,0,0,0,45,45,0,0,26,0,45],[1,0,0,0,0,63,0,0,0,46,1,0,0,26,0,45],[1,0,0,0,0,7,0,79,0,14,0,34,0,50,1,102] >;

C₂×C₉.₄He₃ in GAP, Magma, Sage, TeX

C_2\times C_9._4{\rm He}_3

% in TeX

G:=Group("C2xC9.4He3");

// GroupNames label

G:=SmallGroup(486,76);

// by ID

G=gap.SmallGroup(486,76);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,2169,118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=c^3=d^3=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c*d^-1,e*d*e^-1=b^6*d>;

// generators/relations

G = C2×C9.4He3 order 486 = 2·35

Direct product of C2 and C9.4He3

G = C₂×C₉.₄He₃ order 486 = 2·3⁵

Direct product of C₂ and C₉.₄He₃