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## G = C22×S3×D9order 432 = 24·33

### Direct product of C22, S3 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C22×S3×D9
 Chief series C1 — C3 — C32 — C3×C9 — S3×C9 — S3×D9 — C2×S3×D9 — C22×S3×D9
 Lower central C3×C9 — C22×S3×D9
 Upper central C1 — C22

Generators and relations for C22×S3×D9
G = < a,b,c,d,e,f | a2=b2=c3=d2=e9=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2812 in 434 conjugacy classes, 125 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, S3, C6, C6, C23, C9, C9, C32, D6, D6, C2×C6, C2×C6, C24, D9, D9, C18, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C22×S3, C22×S3, C22×C6, C3×C9, D18, D18, C2×C18, C2×C18, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×C23, C3×D9, S3×C9, C9⋊S3, C3×C18, C22×D9, C22×D9, C22×C18, C2×S32, S3×C2×C6, S3×C2×C6, C22×C3⋊S3, S3×D9, C6×D9, S3×C18, C2×C9⋊S3, C6×C18, C23×D9, C22×S32, C2×S3×D9, C2×C6×D9, S3×C2×C18, C22×C9⋊S3, C22×S3×D9
Quotients: C1, C2, C22, S3, C23, D6, C24, D9, C22×S3, D18, S32, S3×C23, C22×D9, C2×S32, S3×D9, C23×D9, C22×S32, C2×S3×D9, C22×S3×D9

Smallest permutation representation of C22×S3×D9
On 72 points
Generators in S72
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(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)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 45)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 72)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 63)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 57)(2 56)(3 55)(4 63)(5 62)(6 61)(7 60)(8 59)(9 58)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 72)(18 71)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 45)(27 44)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 54)(36 53)

G:=sub<Sym(72)| (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(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)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,46)(8,47)(9,48)(10,45)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,63)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,57)(2,56)(3,55)(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,72)(18,71)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,45)(27,44)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,54)(36,53)>;

G:=Group( (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(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)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,46)(8,47)(9,48)(10,45)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,63)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62), (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,57)(2,56)(3,55)(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,72)(18,71)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,37)(26,45)(27,44)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,54)(36,53) );

G=PermutationGroup([[(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(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),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,46),(8,47),(9,48),(10,45),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,72),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,63),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62)], [(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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,57),(2,56),(3,55),(4,63),(5,62),(6,61),(7,60),(8,59),(9,58),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,72),(18,71),(19,43),(20,42),(21,41),(22,40),(23,39),(24,38),(25,37),(26,45),(27,44),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,54),(36,53)]])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 3C 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 6Q 9A 9B 9C 9D 9E 9F 18A ··· 18I 18J ··· 18R 18S ··· 18AD order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 6 ··· 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 1 1 3 3 3 3 9 9 9 9 27 27 27 27 2 2 4 2 ··· 2 4 4 4 6 6 6 6 18 18 18 18 2 2 2 4 4 4 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D9 D18 D18 S32 C2×S32 S3×D9 C2×S3×D9 kernel C22×S3×D9 C2×S3×D9 C2×C6×D9 S3×C2×C18 C22×C9⋊S3 C22×D9 S3×C2×C6 D18 C2×C18 S3×C6 C62 C22×S3 D6 C2×C6 C2×C6 C6 C22 C2 # reps 1 12 1 1 1 1 1 6 1 6 1 3 18 3 1 3 3 9

Matrix representation of C22×S3×D9 in GL4(𝔽19) generated by

 18 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 0 1 0 0 18 18 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 18 18 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 1 0 0 0 0 17 7 0 0 12 5
,
 1 0 0 0 0 1 0 0 0 0 5 2 0 0 7 14
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,18,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[1,18,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,17,12,0,0,7,5],[1,0,0,0,0,1,0,0,0,0,5,7,0,0,2,14] >;

C22×S3×D9 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times D_9
% in TeX

G:=Group("C2^2xS3xD9");
// GroupNames label

G:=SmallGroup(432,544);
// by ID

G=gap.SmallGroup(432,544);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,3091,662,4037,7069]);
// Polycyclic

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

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