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## G = C4×C9.A4order 432 = 24·33

### Direct product of C4 and C9.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C4×C9.A4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C22×C18 — C2×C9.A4 — C4×C9.A4
 Lower central C22 — C4×C9.A4
 Upper central C1 — C36

Generators and relations for C4×C9.A4
G = < a,b,c,d,e | a4=b9=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

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

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C4 C6 C9 C12 C18 C27 C36 C54 C108 A4 C2×A4 C3.A4 C4×A4 C2×C3.A4 C9.A4 C4×C3.A4 C2×C9.A4 C4×C9.A4 kernel C4×C9.A4 C2×C9.A4 C22×C36 C9.A4 C22×C18 C22×C12 C2×C18 C22×C6 C22×C4 C2×C6 C23 C22 C36 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 18 12 18 36 1 1 2 2 2 6 4 6 12

Matrix representation of C4×C9.A4 in GL4(𝔽109) generated by

 76 0 0 0 0 33 0 0 0 0 33 0 0 0 0 33
,
 27 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 108 0 0 68 0 108
,
 1 0 0 0 0 108 0 0 0 0 108 0 0 41 11 1
,
 3 0 0 0 0 0 1 0 0 68 98 107 0 7 40 11
G:=sub<GL(4,GF(109))| [76,0,0,0,0,33,0,0,0,0,33,0,0,0,0,33],[27,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,68,0,0,108,0,0,0,0,108],[1,0,0,0,0,108,0,41,0,0,108,11,0,0,0,1],[3,0,0,0,0,0,68,7,0,1,98,40,0,0,107,11] >;

C4×C9.A4 in GAP, Magma, Sage, TeX

C_4\times C_9.A_4
% in TeX

G:=Group("C4xC9.A4");
// GroupNames label

G:=SmallGroup(432,40);
// by ID

G=gap.SmallGroup(432,40);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-3,-2,2,42,92,108,4548,7951]);
// Polycyclic

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

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