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## G = C22×C9.A4order 432 = 24·33

### Direct product of C22 and C9.A4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×C9.A4
G = < a,b,c,d,e,f | a2=b2=c9=d2=e2=1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 226 in 97 conjugacy classes, 35 normal (14 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C23, C9, C2×C6, C2×C6, C24, C18, C18, C22×C6, C22×C6, C27, C2×C18, C2×C18, C23×C6, C54, C22×C18, C22×C18, C9.A4, C2×C54, C23×C18, C2×C9.A4, C22×C9.A4
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C27, C3.A4, C2×C18, C22×A4, C54, C2×C3.A4, C9.A4, C2×C54, C22×C3.A4, C2×C9.A4, C22×C9.A4

Smallest permutation representation of C22×C9.A4
On 108 points
Generators in S108
(1 77)(2 78)(3 79)(4 80)(5 81)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 108)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)
(1 51)(2 52)(3 53)(4 54)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)
(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)(55 58 61 64 67 70 73 76 79)(56 59 62 65 68 71 74 77 80)(57 60 63 66 69 72 75 78 81)(82 85 88 91 94 97 100 103 106)(83 86 89 92 95 98 101 104 107)(84 87 90 93 96 99 102 105 108)
(2 78)(3 79)(5 81)(6 55)(8 57)(9 58)(11 60)(12 61)(14 63)(15 64)(17 66)(18 67)(20 69)(21 70)(23 72)(24 73)(26 75)(27 76)(28 108)(29 82)(31 84)(32 85)(34 87)(35 88)(37 90)(38 91)(40 93)(41 94)(43 96)(44 97)(46 99)(47 100)(49 102)(50 103)(52 105)(53 106)
(1 77)(3 79)(4 80)(6 55)(7 56)(9 58)(10 59)(12 61)(13 62)(15 64)(16 65)(18 67)(19 68)(21 70)(22 71)(24 73)(25 74)(27 76)(29 82)(30 83)(32 85)(33 86)(35 88)(36 89)(38 91)(39 92)(41 94)(42 95)(44 97)(45 98)(47 100)(48 101)(50 103)(51 104)(53 106)(54 107)
(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 73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)

G:=sub<Sym(108)| (1,77)(2,78)(3,79)(4,80)(5,81)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,108)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107), (1,51)(2,52)(3,53)(4,54)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (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)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108), (2,78)(3,79)(5,81)(6,55)(8,57)(9,58)(11,60)(12,61)(14,63)(15,64)(17,66)(18,67)(20,69)(21,70)(23,72)(24,73)(26,75)(27,76)(28,108)(29,82)(31,84)(32,85)(34,87)(35,88)(37,90)(38,91)(40,93)(41,94)(43,96)(44,97)(46,99)(47,100)(49,102)(50,103)(52,105)(53,106), (1,77)(3,79)(4,80)(6,55)(7,56)(9,58)(10,59)(12,61)(13,62)(15,64)(16,65)(18,67)(19,68)(21,70)(22,71)(24,73)(25,74)(27,76)(29,82)(30,83)(32,85)(33,86)(35,88)(36,89)(38,91)(39,92)(41,94)(42,95)(44,97)(45,98)(47,100)(48,101)(50,103)(51,104)(53,106)(54,107), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)>;

G:=Group( (1,77)(2,78)(3,79)(4,80)(5,81)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,108)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107), (1,51)(2,52)(3,53)(4,54)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,101)(75,102)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108), (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)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108), (2,78)(3,79)(5,81)(6,55)(8,57)(9,58)(11,60)(12,61)(14,63)(15,64)(17,66)(18,67)(20,69)(21,70)(23,72)(24,73)(26,75)(27,76)(28,108)(29,82)(31,84)(32,85)(34,87)(35,88)(37,90)(38,91)(40,93)(41,94)(43,96)(44,97)(46,99)(47,100)(49,102)(50,103)(52,105)(53,106), (1,77)(3,79)(4,80)(6,55)(7,56)(9,58)(10,59)(12,61)(13,62)(15,64)(16,65)(18,67)(19,68)(21,70)(22,71)(24,73)(25,74)(27,76)(29,82)(30,83)(32,85)(33,86)(35,88)(36,89)(38,91)(39,92)(41,94)(42,95)(44,97)(45,98)(47,100)(48,101)(50,103)(51,104)(53,106)(54,107), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108) );

G=PermutationGroup([[(1,77),(2,78),(3,79),(4,80),(5,81),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,108),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(46,99),(47,100),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107)], [(1,51),(2,52),(3,53),(4,54),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,101),(75,102),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108)], [(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),(55,58,61,64,67,70,73,76,79),(56,59,62,65,68,71,74,77,80),(57,60,63,66,69,72,75,78,81),(82,85,88,91,94,97,100,103,106),(83,86,89,92,95,98,101,104,107),(84,87,90,93,96,99,102,105,108)], [(2,78),(3,79),(5,81),(6,55),(8,57),(9,58),(11,60),(12,61),(14,63),(15,64),(17,66),(18,67),(20,69),(21,70),(23,72),(24,73),(26,75),(27,76),(28,108),(29,82),(31,84),(32,85),(34,87),(35,88),(37,90),(38,91),(40,93),(41,94),(43,96),(44,97),(46,99),(47,100),(49,102),(50,103),(52,105),(53,106)], [(1,77),(3,79),(4,80),(6,55),(7,56),(9,58),(10,59),(12,61),(13,62),(15,64),(16,65),(18,67),(19,68),(21,70),(22,71),(24,73),(25,74),(27,76),(29,82),(30,83),(32,85),(33,86),(35,88),(36,89),(38,91),(39,92),(41,94),(42,95),(44,97),(45,98),(47,100),(48,101),(50,103),(51,104),(53,106),(54,107)], [(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),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)]])

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A ··· 6F 6G ··· 6N 9A ··· 9F 18A ··· 18R 18S ··· 18AP 27A ··· 27R 54A ··· 54BB order 1 2 2 2 2 2 2 2 3 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 1 1 3 3 3 3 1 1 1 ··· 1 3 ··· 3 1 ··· 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C6 C9 C18 C27 C54 A4 C2×A4 C3.A4 C2×C3.A4 C9.A4 C2×C9.A4 kernel C22×C9.A4 C2×C9.A4 C23×C18 C22×C18 C23×C6 C22×C6 C24 C23 C2×C18 C18 C2×C6 C6 C22 C2 # reps 1 3 2 6 6 18 18 54 1 3 2 6 6 18

Matrix representation of C22×C9.A4 in GL4(𝔽109) generated by

 108 0 0 0 0 108 0 0 0 0 108 0 0 0 0 108
,
 1 0 0 0 0 108 0 0 0 0 108 0 0 0 0 108
,
 1 0 0 0 0 38 0 0 0 0 38 0 0 0 0 38
,
 1 0 0 0 0 1 0 0 0 0 108 0 0 0 0 108
,
 1 0 0 0 0 108 0 0 0 0 108 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 38 0 0
G:=sub<GL(4,GF(109))| [108,0,0,0,0,108,0,0,0,0,108,0,0,0,0,108],[1,0,0,0,0,108,0,0,0,0,108,0,0,0,0,108],[1,0,0,0,0,38,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,1,0,0,0,0,108,0,0,0,0,108],[1,0,0,0,0,108,0,0,0,0,108,0,0,0,0,1],[1,0,0,0,0,0,0,38,0,1,0,0,0,0,1,0] >;

C22×C9.A4 in GAP, Magma, Sage, TeX

C_2^2\times C_9.A_4
% in TeX

G:=Group("C2^2xC9.A4");
// GroupNames label

G:=SmallGroup(432,225);
// by ID

G=gap.SmallGroup(432,225);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,79,108,2287,3989]);
// Polycyclic

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

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