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## G = A4×C3×C12order 432 = 24·33

### Direct product of C3×C12 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C3×C12
 Chief series C1 — C22 — C23 — C22×C6 — C2×C62 — A4×C3×C6 — A4×C3×C12
 Lower central C22 — A4×C3×C12
 Upper central C1 — C3×C12

Generators and relations for A4×C3×C12
G = < a,b,c,d,e | a3=b12=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 492 in 210 conjugacy classes, 102 normal (15 characteristic)
C1, C2, C2 [×2], C3 [×4], C3 [×9], C4, C4, C22, C22 [×2], C6 [×4], C6 [×17], C2×C4 [×2], C23, C32, C32 [×12], C12 [×4], C12 [×13], A4 [×9], C2×C6 [×4], C2×C6 [×8], C22×C4, C3×C6, C3×C6 [×14], C2×C12 [×8], C2×A4 [×9], C22×C6 [×4], C33, C3×C12, C3×C12 [×13], C3×A4 [×12], C62, C62 [×2], C4×A4 [×9], C22×C12 [×4], C32×C6, C6×C12 [×2], C6×A4 [×12], C2×C62, C32×C12, C32×A4, C12×A4 [×12], C2×C6×C12, A4×C3×C6, A4×C3×C12
Quotients: C1, C2, C3 [×13], C4, C6 [×13], C32 [×13], C12 [×13], A4, C3×C6 [×13], C2×A4, C33, C3×C12 [×13], C3×A4 [×4], C4×A4, C32×C6, C6×A4 [×4], C32×C12, C32×A4, C12×A4 [×4], A4×C3×C6, A4×C3×C12

Smallest permutation representation of A4×C3×C12
On 108 points
Generators in S108
(1 39 15)(2 40 16)(3 41 17)(4 42 18)(5 43 19)(6 44 20)(7 45 21)(8 46 22)(9 47 23)(10 48 24)(11 37 13)(12 38 14)(25 102 54)(26 103 55)(27 104 56)(28 105 57)(29 106 58)(30 107 59)(31 108 60)(32 97 49)(33 98 50)(34 99 51)(35 100 52)(36 101 53)(61 75 85)(62 76 86)(63 77 87)(64 78 88)(65 79 89)(66 80 90)(67 81 91)(68 82 92)(69 83 93)(70 84 94)(71 73 95)(72 74 96)
(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)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 55 87)(2 56 88)(3 57 89)(4 58 90)(5 59 91)(6 60 92)(7 49 93)(8 50 94)(9 51 95)(10 52 96)(11 53 85)(12 54 86)(13 101 75)(14 102 76)(15 103 77)(16 104 78)(17 105 79)(18 106 80)(19 107 81)(20 108 82)(21 97 83)(22 98 84)(23 99 73)(24 100 74)(25 62 38)(26 63 39)(27 64 40)(28 65 41)(29 66 42)(30 67 43)(31 68 44)(32 69 45)(33 70 46)(34 71 47)(35 72 48)(36 61 37)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Z 4A 4B 4C 4D 6A ··· 6H 6I ··· 6X 6Y ··· 6AP 12A ··· 12P 12Q ··· 12AF 12AG ··· 12BP order 1 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 3 3 1 ··· 1 4 ··· 4 1 1 3 3 1 ··· 1 3 ··· 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C4 C6 C6 C12 C12 A4 C2×A4 C3×A4 C4×A4 C6×A4 C12×A4 kernel A4×C3×C12 A4×C3×C6 C12×A4 C2×C6×C12 C32×A4 C6×A4 C2×C62 C3×A4 C62 C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 24 2 2 24 2 48 4 1 1 8 2 8 16

Matrix representation of A4×C3×C12 in GL4(𝔽13) generated by

 3 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 11 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 12 0 0 10 0 12
,
 1 0 0 0 0 12 0 0 0 0 12 0 0 3 9 1
,
 9 0 0 0 0 0 9 0 0 12 10 8 0 0 0 3
G:=sub<GL(4,GF(13))| [3,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[11,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,10,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,3,0,0,12,9,0,0,0,1],[9,0,0,0,0,0,12,0,0,9,10,0,0,0,8,3] >;

A4×C3×C12 in GAP, Magma, Sage, TeX

A_4\times C_3\times C_{12}
% in TeX

G:=Group("A4xC3xC12");
// GroupNames label

G:=SmallGroup(432,697);
// by ID

G=gap.SmallGroup(432,697);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,-2,2,378,4548,7951]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^12=c^2=d^2=e^3=1,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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