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## G = C9×A4⋊C4order 432 = 24·33

### Direct product of C9 and A4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C9×A4⋊C4
 Chief series C1 — C22 — A4 — C3×A4 — C6×A4 — A4×C18 — C9×A4⋊C4
 Lower central A4 — C9×A4⋊C4
 Upper central C1 — C18

Generators and relations for C9×A4⋊C4
G = < a,b,c,d,e | a9=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Subgroups: 196 in 65 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, A4, A4, C2×C6, C2×C6, C22⋊C4, C18, C18, C3×C6, C2×C12, C2×A4, C2×A4, C22×C6, C3×C9, C36, C3.A4, C2×C18, C2×C18, C3×Dic3, C3×A4, C3×C22⋊C4, A4⋊C4, C3×C18, C2×C36, C2×C3.A4, C22×C18, C6×A4, C9×Dic3, C9×A4, C9×C22⋊C4, C3×A4⋊C4, A4×C18, C9×A4⋊C4
Quotients: C1, C2, C3, C4, S3, C6, C9, Dic3, C12, C18, C3×S3, S4, C36, C3×Dic3, A4⋊C4, S3×C9, C3×S4, C9×Dic3, C3×A4⋊C4, C9×S4, C9×A4⋊C4

Smallest permutation representation of C9×A4⋊C4
On 108 points
Generators in S108
(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)
(10 35)(11 36)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 100)(55 86)(56 87)(57 88)(58 89)(59 90)(60 82)(61 83)(62 84)(63 85)(73 96)(74 97)(75 98)(76 99)(77 91)(78 92)(79 93)(80 94)(81 95)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 64)(7 65)(8 66)(9 67)(10 35)(11 36)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 50)(20 51)(21 52)(22 53)(23 54)(24 46)(25 47)(26 48)(27 49)(55 86)(56 87)(57 88)(58 89)(59 90)(60 82)(61 83)(62 84)(63 85)
(1 74 58)(2 75 59)(3 76 60)(4 77 61)(5 78 62)(6 79 63)(7 80 55)(8 81 56)(9 73 57)(10 25 41)(11 26 42)(12 27 43)(13 19 44)(14 20 45)(15 21 37)(16 22 38)(17 23 39)(18 24 40)(28 49 107)(29 50 108)(30 51 100)(31 52 101)(32 53 102)(33 54 103)(34 46 104)(35 47 105)(36 48 106)(64 93 85)(65 94 86)(66 95 87)(67 96 88)(68 97 89)(69 98 90)(70 99 82)(71 91 83)(72 92 84)
(1 36 68 11)(2 28 69 12)(3 29 70 13)(4 30 71 14)(5 31 72 15)(6 32 64 16)(7 33 65 17)(8 34 66 18)(9 35 67 10)(19 60 50 82)(20 61 51 83)(21 62 52 84)(22 63 53 85)(23 55 54 86)(24 56 46 87)(25 57 47 88)(26 58 48 89)(27 59 49 90)(37 78 101 92)(38 79 102 93)(39 80 103 94)(40 81 104 95)(41 73 105 96)(42 74 106 97)(43 75 107 98)(44 76 108 99)(45 77 100 91)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9F 9G ··· 9L 12A ··· 12H 18A ··· 18F 18G ··· 18R 18S ··· 18X 36A ··· 36X order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 3 3 1 1 8 8 8 6 6 6 6 1 1 3 3 3 3 8 8 8 1 ··· 1 8 ··· 8 6 ··· 6 1 ··· 1 3 ··· 3 8 ··· 8 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 type + + + - + image C1 C2 C3 C4 C6 C9 C12 C18 C36 S3 Dic3 C3×S3 C3×Dic3 S3×C9 C9×Dic3 S4 A4⋊C4 C3×S4 C3×A4⋊C4 C9×S4 C9×A4⋊C4 kernel C9×A4⋊C4 A4×C18 C3×A4⋊C4 C9×A4 C6×A4 A4⋊C4 C3×A4 C2×A4 A4 C22×C18 C2×C18 C22×C6 C2×C6 C23 C22 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 1 1 2 2 6 6 2 2 4 4 12 12

Matrix representation of C9×A4⋊C4 in GL3(𝔽37) generated by

 16 0 0 0 16 0 0 0 16
,
 1 0 0 0 36 0 0 0 36
,
 36 0 0 0 1 0 0 0 36
,
 0 0 1 1 0 0 0 1 0
,
 0 0 6 0 6 0 6 0 0
G:=sub<GL(3,GF(37))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,36,0,0,0,36],[36,0,0,0,1,0,0,0,36],[0,1,0,0,0,1,1,0,0],[0,0,6,0,6,0,6,0,0] >;

C9×A4⋊C4 in GAP, Magma, Sage, TeX

C_9\times A_4\rtimes C_4
% in TeX

G:=Group("C9xA4:C4");
// GroupNames label

G:=SmallGroup(432,242);
// by ID

G=gap.SmallGroup(432,242);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-3,-2,2,42,92,2524,9077,285,5298,475]);
// Polycyclic

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

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