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## G = C9×C4.A4order 432 = 24·33

### Direct product of C9 and C4.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C9×C4.A4
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C9 — C9×SL2(𝔽3) — C9×C4.A4
 Lower central Q8 — C9×C4.A4
 Upper central C1 — C36

Generators and relations for C9×C4.A4
G = < a,b,c,d,e | a9=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

Subgroups: 167 in 65 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C3 [×3], C4, C4, C22, C6, C6 [×4], C2×C4, D4, Q8, C9, C9 [×2], C32, C12, C12 [×4], C2×C6, C4○D4, C18, C18 [×3], C3×C6, SL2(𝔽3) [×3], C2×C12, C3×D4, C3×Q8, C3×C9, C36, C36 [×3], C2×C18, C3×C12, C4.A4 [×3], C3×C4○D4, C3×C18, Q8⋊C9 [×2], C2×C36, D4×C9, Q8×C9, C3×SL2(𝔽3), C3×C36, Q8.C18 [×2], C9×C4○D4, C3×C4.A4, C9×SL2(𝔽3), C9×C4.A4
Quotients: C1, C2, C3 [×4], C6 [×4], C9 [×3], C32, A4, C18 [×3], C3×C6, C2×A4, C3×C9, C3×A4, C4.A4, C3×C18, C6×A4, C9×A4, C3×C4.A4, A4×C18, C9×C4.A4

Smallest permutation representation of C9×C4.A4
On 144 points
Generators in S144
(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)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144)
(1 63 109 138)(2 55 110 139)(3 56 111 140)(4 57 112 141)(5 58 113 142)(6 59 114 143)(7 60 115 144)(8 61 116 136)(9 62 117 137)(10 73 21 133)(11 74 22 134)(12 75 23 135)(13 76 24 127)(14 77 25 128)(15 78 26 129)(16 79 27 130)(17 80 19 131)(18 81 20 132)(28 83 72 118)(29 84 64 119)(30 85 65 120)(31 86 66 121)(32 87 67 122)(33 88 68 123)(34 89 69 124)(35 90 70 125)(36 82 71 126)(37 92 49 104)(38 93 50 105)(39 94 51 106)(40 95 52 107)(41 96 53 108)(42 97 54 100)(43 98 46 101)(44 99 47 102)(45 91 48 103)
(1 119 109 84)(2 120 110 85)(3 121 111 86)(4 122 112 87)(5 123 113 88)(6 124 114 89)(7 125 115 90)(8 126 116 82)(9 118 117 83)(10 40 21 52)(11 41 22 53)(12 42 23 54)(13 43 24 46)(14 44 25 47)(15 45 26 48)(16 37 27 49)(17 38 19 50)(18 39 20 51)(28 137 72 62)(29 138 64 63)(30 139 65 55)(31 140 66 56)(32 141 67 57)(33 142 68 58)(34 143 69 59)(35 144 70 60)(36 136 71 61)(73 95 133 107)(74 96 134 108)(75 97 135 100)(76 98 127 101)(77 99 128 102)(78 91 129 103)(79 92 130 104)(80 93 131 105)(81 94 132 106)
(1 132 109 81)(2 133 110 73)(3 134 111 74)(4 135 112 75)(5 127 113 76)(6 128 114 77)(7 129 115 78)(8 130 116 79)(9 131 117 80)(10 139 21 55)(11 140 22 56)(12 141 23 57)(13 142 24 58)(14 143 25 59)(15 144 26 60)(16 136 27 61)(17 137 19 62)(18 138 20 63)(28 50 72 38)(29 51 64 39)(30 52 65 40)(31 53 66 41)(32 54 67 42)(33 46 68 43)(34 47 69 44)(35 48 70 45)(36 49 71 37)(82 104 126 92)(83 105 118 93)(84 106 119 94)(85 107 120 95)(86 108 121 96)(87 100 122 97)(88 101 123 98)(89 102 124 99)(90 103 125 91)
(1 4 7)(2 5 8)(3 6 9)(10 43 36)(11 44 28)(12 45 29)(13 37 30)(14 38 31)(15 39 32)(16 40 33)(17 41 34)(18 42 35)(19 53 69)(20 54 70)(21 46 71)(22 47 72)(23 48 64)(24 49 65)(25 50 66)(26 51 67)(27 52 68)(55 58 61)(56 59 62)(57 60 63)(73 98 82)(74 99 83)(75 91 84)(76 92 85)(77 93 86)(78 94 87)(79 95 88)(80 96 89)(81 97 90)(100 125 132)(101 126 133)(102 118 134)(103 119 135)(104 120 127)(105 121 128)(106 122 129)(107 123 130)(108 124 131)(109 112 115)(110 113 116)(111 114 117)(136 139 142)(137 140 143)(138 141 144)

G:=sub<Sym(144)| (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)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,63,109,138)(2,55,110,139)(3,56,111,140)(4,57,112,141)(5,58,113,142)(6,59,114,143)(7,60,115,144)(8,61,116,136)(9,62,117,137)(10,73,21,133)(11,74,22,134)(12,75,23,135)(13,76,24,127)(14,77,25,128)(15,78,26,129)(16,79,27,130)(17,80,19,131)(18,81,20,132)(28,83,72,118)(29,84,64,119)(30,85,65,120)(31,86,66,121)(32,87,67,122)(33,88,68,123)(34,89,69,124)(35,90,70,125)(36,82,71,126)(37,92,49,104)(38,93,50,105)(39,94,51,106)(40,95,52,107)(41,96,53,108)(42,97,54,100)(43,98,46,101)(44,99,47,102)(45,91,48,103), (1,119,109,84)(2,120,110,85)(3,121,111,86)(4,122,112,87)(5,123,113,88)(6,124,114,89)(7,125,115,90)(8,126,116,82)(9,118,117,83)(10,40,21,52)(11,41,22,53)(12,42,23,54)(13,43,24,46)(14,44,25,47)(15,45,26,48)(16,37,27,49)(17,38,19,50)(18,39,20,51)(28,137,72,62)(29,138,64,63)(30,139,65,55)(31,140,66,56)(32,141,67,57)(33,142,68,58)(34,143,69,59)(35,144,70,60)(36,136,71,61)(73,95,133,107)(74,96,134,108)(75,97,135,100)(76,98,127,101)(77,99,128,102)(78,91,129,103)(79,92,130,104)(80,93,131,105)(81,94,132,106), (1,132,109,81)(2,133,110,73)(3,134,111,74)(4,135,112,75)(5,127,113,76)(6,128,114,77)(7,129,115,78)(8,130,116,79)(9,131,117,80)(10,139,21,55)(11,140,22,56)(12,141,23,57)(13,142,24,58)(14,143,25,59)(15,144,26,60)(16,136,27,61)(17,137,19,62)(18,138,20,63)(28,50,72,38)(29,51,64,39)(30,52,65,40)(31,53,66,41)(32,54,67,42)(33,46,68,43)(34,47,69,44)(35,48,70,45)(36,49,71,37)(82,104,126,92)(83,105,118,93)(84,106,119,94)(85,107,120,95)(86,108,121,96)(87,100,122,97)(88,101,123,98)(89,102,124,99)(90,103,125,91), (1,4,7)(2,5,8)(3,6,9)(10,43,36)(11,44,28)(12,45,29)(13,37,30)(14,38,31)(15,39,32)(16,40,33)(17,41,34)(18,42,35)(19,53,69)(20,54,70)(21,46,71)(22,47,72)(23,48,64)(24,49,65)(25,50,66)(26,51,67)(27,52,68)(55,58,61)(56,59,62)(57,60,63)(73,98,82)(74,99,83)(75,91,84)(76,92,85)(77,93,86)(78,94,87)(79,95,88)(80,96,89)(81,97,90)(100,125,132)(101,126,133)(102,118,134)(103,119,135)(104,120,127)(105,121,128)(106,122,129)(107,123,130)(108,124,131)(109,112,115)(110,113,116)(111,114,117)(136,139,142)(137,140,143)(138,141,144)>;

G:=Group( (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)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,63,109,138)(2,55,110,139)(3,56,111,140)(4,57,112,141)(5,58,113,142)(6,59,114,143)(7,60,115,144)(8,61,116,136)(9,62,117,137)(10,73,21,133)(11,74,22,134)(12,75,23,135)(13,76,24,127)(14,77,25,128)(15,78,26,129)(16,79,27,130)(17,80,19,131)(18,81,20,132)(28,83,72,118)(29,84,64,119)(30,85,65,120)(31,86,66,121)(32,87,67,122)(33,88,68,123)(34,89,69,124)(35,90,70,125)(36,82,71,126)(37,92,49,104)(38,93,50,105)(39,94,51,106)(40,95,52,107)(41,96,53,108)(42,97,54,100)(43,98,46,101)(44,99,47,102)(45,91,48,103), (1,119,109,84)(2,120,110,85)(3,121,111,86)(4,122,112,87)(5,123,113,88)(6,124,114,89)(7,125,115,90)(8,126,116,82)(9,118,117,83)(10,40,21,52)(11,41,22,53)(12,42,23,54)(13,43,24,46)(14,44,25,47)(15,45,26,48)(16,37,27,49)(17,38,19,50)(18,39,20,51)(28,137,72,62)(29,138,64,63)(30,139,65,55)(31,140,66,56)(32,141,67,57)(33,142,68,58)(34,143,69,59)(35,144,70,60)(36,136,71,61)(73,95,133,107)(74,96,134,108)(75,97,135,100)(76,98,127,101)(77,99,128,102)(78,91,129,103)(79,92,130,104)(80,93,131,105)(81,94,132,106), (1,132,109,81)(2,133,110,73)(3,134,111,74)(4,135,112,75)(5,127,113,76)(6,128,114,77)(7,129,115,78)(8,130,116,79)(9,131,117,80)(10,139,21,55)(11,140,22,56)(12,141,23,57)(13,142,24,58)(14,143,25,59)(15,144,26,60)(16,136,27,61)(17,137,19,62)(18,138,20,63)(28,50,72,38)(29,51,64,39)(30,52,65,40)(31,53,66,41)(32,54,67,42)(33,46,68,43)(34,47,69,44)(35,48,70,45)(36,49,71,37)(82,104,126,92)(83,105,118,93)(84,106,119,94)(85,107,120,95)(86,108,121,96)(87,100,122,97)(88,101,123,98)(89,102,124,99)(90,103,125,91), (1,4,7)(2,5,8)(3,6,9)(10,43,36)(11,44,28)(12,45,29)(13,37,30)(14,38,31)(15,39,32)(16,40,33)(17,41,34)(18,42,35)(19,53,69)(20,54,70)(21,46,71)(22,47,72)(23,48,64)(24,49,65)(25,50,66)(26,51,67)(27,52,68)(55,58,61)(56,59,62)(57,60,63)(73,98,82)(74,99,83)(75,91,84)(76,92,85)(77,93,86)(78,94,87)(79,95,88)(80,96,89)(81,97,90)(100,125,132)(101,126,133)(102,118,134)(103,119,135)(104,120,127)(105,121,128)(106,122,129)(107,123,130)(108,124,131)(109,112,115)(110,113,116)(111,114,117)(136,139,142)(137,140,143)(138,141,144) );

G=PermutationGroup([(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),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144)], [(1,63,109,138),(2,55,110,139),(3,56,111,140),(4,57,112,141),(5,58,113,142),(6,59,114,143),(7,60,115,144),(8,61,116,136),(9,62,117,137),(10,73,21,133),(11,74,22,134),(12,75,23,135),(13,76,24,127),(14,77,25,128),(15,78,26,129),(16,79,27,130),(17,80,19,131),(18,81,20,132),(28,83,72,118),(29,84,64,119),(30,85,65,120),(31,86,66,121),(32,87,67,122),(33,88,68,123),(34,89,69,124),(35,90,70,125),(36,82,71,126),(37,92,49,104),(38,93,50,105),(39,94,51,106),(40,95,52,107),(41,96,53,108),(42,97,54,100),(43,98,46,101),(44,99,47,102),(45,91,48,103)], [(1,119,109,84),(2,120,110,85),(3,121,111,86),(4,122,112,87),(5,123,113,88),(6,124,114,89),(7,125,115,90),(8,126,116,82),(9,118,117,83),(10,40,21,52),(11,41,22,53),(12,42,23,54),(13,43,24,46),(14,44,25,47),(15,45,26,48),(16,37,27,49),(17,38,19,50),(18,39,20,51),(28,137,72,62),(29,138,64,63),(30,139,65,55),(31,140,66,56),(32,141,67,57),(33,142,68,58),(34,143,69,59),(35,144,70,60),(36,136,71,61),(73,95,133,107),(74,96,134,108),(75,97,135,100),(76,98,127,101),(77,99,128,102),(78,91,129,103),(79,92,130,104),(80,93,131,105),(81,94,132,106)], [(1,132,109,81),(2,133,110,73),(3,134,111,74),(4,135,112,75),(5,127,113,76),(6,128,114,77),(7,129,115,78),(8,130,116,79),(9,131,117,80),(10,139,21,55),(11,140,22,56),(12,141,23,57),(13,142,24,58),(14,143,25,59),(15,144,26,60),(16,136,27,61),(17,137,19,62),(18,138,20,63),(28,50,72,38),(29,51,64,39),(30,52,65,40),(31,53,66,41),(32,54,67,42),(33,46,68,43),(34,47,69,44),(35,48,70,45),(36,49,71,37),(82,104,126,92),(83,105,118,93),(84,106,119,94),(85,107,120,95),(86,108,121,96),(87,100,122,97),(88,101,123,98),(89,102,124,99),(90,103,125,91)], [(1,4,7),(2,5,8),(3,6,9),(10,43,36),(11,44,28),(12,45,29),(13,37,30),(14,38,31),(15,39,32),(16,40,33),(17,41,34),(18,42,35),(19,53,69),(20,54,70),(21,46,71),(22,47,72),(23,48,64),(24,49,65),(25,50,66),(26,51,67),(27,52,68),(55,58,61),(56,59,62),(57,60,63),(73,98,82),(74,99,83),(75,91,84),(76,92,85),(77,93,86),(78,94,87),(79,95,88),(80,96,89),(81,97,90),(100,125,132),(101,126,133),(102,118,134),(103,119,135),(104,120,127),(105,121,128),(106,122,129),(107,123,130),(108,124,131),(109,112,115),(110,113,116),(111,114,117),(136,139,142),(137,140,143),(138,141,144)])

126 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3H 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 9A ··· 9F 9G ··· 9R 12A 12B 12C 12D 12E ··· 12P 12Q 12R 18A ··· 18F 18G ··· 18R 18S ··· 18X 36A ··· 36L 36M ··· 36AJ 36AK ··· 36AP order 1 2 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 12 ··· 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 6 1 1 4 ··· 4 1 1 6 1 1 4 ··· 4 6 6 1 ··· 1 4 ··· 4 1 1 1 1 4 ··· 4 6 6 1 ··· 1 4 ··· 4 6 ··· 6 1 ··· 1 4 ··· 4 6 ··· 6

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 C4.A4 C3×C4.A4 C9×C4.A4 A4 C2×A4 C3×A4 C6×A4 C9×A4 A4×C18 kernel C9×C4.A4 C9×SL2(𝔽3) Q8.C18 C9×C4○D4 C3×C4.A4 Q8⋊C9 Q8×C9 C3×SL2(𝔽3) C4.A4 SL2(𝔽3) C9 C3 C1 C36 C18 C12 C6 C4 C2 # reps 1 1 4 2 2 4 2 2 18 18 6 12 36 1 1 2 2 6 6

Matrix representation of C9×C4.A4 in GL2(𝔽37) generated by

 9 0 0 9
,
 6 0 0 6
,
 26 10 10 11
,
 0 36 1 0
,
 0 27 26 36
G:=sub<GL(2,GF(37))| [9,0,0,9],[6,0,0,6],[26,10,10,11],[0,1,36,0],[0,26,27,36] >;

C9×C4.A4 in GAP, Magma, Sage, TeX

C_9\times C_4.A_4
% in TeX

G:=Group("C9xC4.A4");
// GroupNames label

G:=SmallGroup(432,329);
// by ID

G=gap.SmallGroup(432,329);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,79,1901,172,3414,285,124]);
// Polycyclic

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

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