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## G = C2×D9⋊A4order 432 = 24·33

### Direct product of C2 and D9⋊A4

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

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