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## G = C2×C9⋊A4order 216 = 23·33

### Direct product of C2 and C9⋊A4

Aliases: C2×C9⋊A4, C18⋊A4, C2313- 1+2, C92(C2×A4), (C6×A4).C3, (C3×A4).C6, (C2×C18)⋊7C6, C3.A42C6, C3.3(C6×A4), C6.4(C3×A4), (C22×C18)⋊2C3, (C22×C6).2C32, C221(C2×3- 1+2), (C2×C3.A4)⋊1C3, (C2×C6).2(C3×C6), SmallGroup(216,104)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C9⋊A4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9⋊A4 — C2×C9⋊A4
 Lower central C22 — C2×C6 — C2×C9⋊A4
 Upper central C1 — C6 — C18

Generators and relations for C2×C9⋊A4
G = < a,b,c,d,e | a2=b9=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b7, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C2×C9⋊A4
On 54 points
Generators in S54
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 19)(9 20)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 46)(17 47)(18 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(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)
(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 46)(17 47)(18 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 19)(9 20)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(1 52 28)(2 47 35)(3 51 33)(4 46 31)(5 50 29)(6 54 36)(7 49 34)(8 53 32)(9 48 30)(10 43 27)(11 38 25)(12 42 23)(13 37 21)(14 41 19)(15 45 26)(16 40 24)(17 44 22)(18 39 20)

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

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

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

C2×C9⋊A4 is a maximal subgroup of   Dic9⋊A4
C2×C9⋊A4 is a maximal quotient of   C36.A4

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D 6E 6F 6G 6H 9A 9B 9C 9D 9E 9F 18A ··· 18N 18O 18P 18Q 18R order 1 2 2 2 3 3 3 3 6 6 6 6 6 6 6 6 9 9 9 9 9 9 18 ··· 18 18 18 18 18 size 1 1 3 3 1 1 12 12 1 1 3 3 3 3 12 12 3 3 12 12 12 12 3 ··· 3 12 12 12 12

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 A4 C2×A4 3- 1+2 C3×A4 C2×3- 1+2 C6×A4 C9⋊A4 C2×C9⋊A4 kernel C2×C9⋊A4 C9⋊A4 C2×C3.A4 C22×C18 C6×A4 C3.A4 C2×C18 C3×A4 C18 C9 C23 C6 C22 C3 C2 C1 # reps 1 1 4 2 2 4 2 2 1 1 2 2 2 2 6 6

Matrix representation of C2×C9⋊A4 in GL3(𝔽19) generated by

 18 0 0 0 18 0 0 0 18
,
 9 0 0 0 6 0 0 0 4
,
 1 0 0 0 18 0 0 0 18
,
 18 0 0 0 18 0 0 0 1
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[9,0,0,0,6,0,0,0,4],[1,0,0,0,18,0,0,0,18],[18,0,0,0,18,0,0,0,1],[0,0,1,1,0,0,0,1,0] >;

C2×C9⋊A4 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes A_4
% in TeX

G:=Group("C2xC9:A4");
// GroupNames label

G:=SmallGroup(216,104);
// by ID

G=gap.SmallGroup(216,104);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,224,68,1630,2927]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^7,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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