C2xD9:A4 - GroupNames

G = C₂×D₉⋊A₄ order 432 = 2⁴·3³

Direct product of C₂ and D₉⋊A₄

direct product, metabelian, soluble, monomial

Aliases: C₂×D₉⋊A₄, D₁₈⋊A₄, C₁₈⋊(C₂×A₄), D₉⋊(C₂×A₄), C₉⋊A₄⋊C₂², C₉⋊(C₂²×A₄), (C₃×A₄).D₆, (C₆×A₄).₄S₃, C₆.₁₀(S₃×A₄), C₂³⋊₂(C₉⋊C₆), (C₂²×C₁₈)⋊₃C₆, (C₂³×D₉)⋊₂C₃, (C₂²×D₉)⋊₄C₆, (C₂×C₉⋊A₄)⋊C₂, C₃.₁(C₂×S₃×A₄), (C₂×C₁₈)⋊₂(C₂×C₆), C₂²⋊₂(C₂×C₉⋊C₆), (C₂×C₆).₅(S₃×C₆), (C₂²×C₆).₁₄(C₃×S₃), SmallGroup(432,539)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₈ — C₂×D₉⋊A₄

Chief series

C₁ — C₃ — C₉ — C₂×C₁₈ — C₉⋊A₄ — D₉⋊A₄ — C₂×D₉⋊A₄

Lower central

C₂×C₁₈ — C₂×D₉⋊A₄

Upper central

C₁ — C₂

Generators and relations for C₂×D₉⋊A₄
G = < a,b,c,d,e,f | a²=b⁹=c²=d²=e²=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b^-1, bd=db, be=eb, fbf^-1=b⁴, cd=dc, ce=ec, fcf^-1=b³c, fdf^-1=de=ed, fef^-1=d >

Subgroups: 976 in 121 conjugacy classes, 25 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², S₃, C₆, C₆, C₂³, C₂³, C₉, C₉, C₃², A₄, D₆, C₂×C₆, C₂×C₆, C₂⁴, D₉, D₉, C₁₈, C₁₈, C₃×S₃, C₃×C₆, C₂×A₄, C₂²×S₃, C₂²×C₆, 3_-¹⁺², C₃.A₄, D₁₈, D₁₈, C₂×C₁₈, C₂×C₁₈, C₃×A₄, S₃×C₆, C₂²×A₄, S₃×C₂³, C₉⋊C₆, C₂×3_-¹⁺², C₂×C₃.A₄, C₂²×D₉, C₂²×D₉, C₂²×C₁₈, S₃×A₄, C₆×A₄, C₉⋊A₄, C₂×C₉⋊C₆, C₂³×D₉, C₂×S₃×A₄, D₉⋊A₄, C₂×C₉⋊A₄, C₂×D₉⋊A₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, A₄, D₆, C₂×C₆, C₃×S₃, C₂×A₄, S₃×C₆, C₂²×A₄, C₉⋊C₆, S₃×A₄, C₂×C₉⋊C₆, C₂×S₃×A₄, D₉⋊A₄, C₂×D₉⋊A₄

Smallest permutation representation of C₂×D₉⋊A₄
►On 54 points

Generators in S₅₄

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

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

(1 17)(2 18)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)

(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	6A	6B	6C	6D	6E	6F	6G	6H	6I	9A	9B	9C	18A	···	18G	18H	18I
order	1	2	2	2	2	2	2	2	3	3	3	6	6	6	6	6	6	6	6	6	9	9	9	18	···	18	18	18
size	1	1	3	3	9	9	27	27	2	12	12	2	6	6	12	12	36	36	36	36	6	24	24	6	···	6	24	24

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	6	6	6	6	6	6
type	+	+	+				+	+			+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃×S₃	S₃×C₆	A₄	C₂×A₄	C₂×A₄	C₉⋊C₆	S₃×A₄	C₂×C₉⋊C₆	C₂×S₃×A₄	D₉⋊A₄	C₂×D₉⋊A₄
kernel	C₂×D₉⋊A₄	D₉⋊A₄	C₂×C₉⋊A₄	C₂³×D₉	C₂²×D₉	C₂²×C₁₈	C₆×A₄	C₃×A₄	C₂²×C₆	C₂×C₆	D₁₈	D₉	C₁₈	C₂³	C₆	C₂²	C₃	C₂	C₁
# reps	1	2	1	2	4	2	1	1	2	2	1	2	1	1	1	1	1	3	3

Matrix representation of C₂×D₉⋊A₄ ►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

12	17	0	0	0	0
2	14	0	0	0	0
0	0	2	14	0	0
0	0	5	7	0	0
0	0	0	0	5	7
0	0	0	0	12	17

7	2	0	0	0	0
14	12	0	0	0	0
0	0	17	5	0	0
0	0	7	2	0	0
0	0	0	0	14	12
0	0	0	0	17	5

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	18	0
0	0	0	0	0	18

1	0	0	0	0	0
0	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	0	0	18

0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1
1	0	0	0	0	0
0	1	0	0	0	0

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[12,2,0,0,0,0,17,14,0,0,0,0,0,0,2,5,0,0,0,0,14,7,0,0,0,0,0,0,5,12,0,0,0,0,7,17],[7,14,0,0,0,0,2,12,0,0,0,0,0,0,17,7,0,0,0,0,5,2,0,0,0,0,0,0,14,17,0,0,0,0,12,5],[18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂×D₉⋊A₄ in GAP, Magma, Sage, TeX

C_2\times D_9\rtimes A_4

% in TeX

G:=Group("C2xD9:A4");

// GroupNames label

G:=SmallGroup(432,539);

// by ID

G=gap.SmallGroup(432,539);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-3,269,123,10085,2035,292,14118]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^9=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=b^4,c*d=d*c,c*e=e*c,f*c*f^-1=b^3*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;

// generators/relations

G = C2×D9⋊A4 order 432 = 24·33

Direct product of C2 and D9⋊A4

G = C₂×D₉⋊A₄ order 432 = 2⁴·3³

Direct product of C₂ and D₉⋊A₄