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G = A4xC3xC12order 432 = 24·33

Direct product of C3xC12 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: A4xC3xC12, C62:12C12, (C22xC4):C33, C6.18(C6xA4), (C6xA4).12C6, (C22xC12):C32, C22:(C32xC12), C23.(C32xC6), (C2xC62).19C6, (C2xC6xC12):3C3, C2.1(A4xC3xC6), (A4xC3xC6).6C2, (C2xC6):2(C3xC12), (C2xA4).2(C3xC6), (C3xC6).25(C2xA4), (C22xC6).12(C3xC6), SmallGroup(432,697)

Series: Derived Chief Lower central Upper central

C1C22 — A4xC3xC12
C1C22C23C22xC6C2xC62A4xC3xC6 — A4xC3xC12
C22 — A4xC3xC12
C1C3xC12

Generators and relations for A4xC3xC12
 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, C3, C3, C4, C4, C22, C22, C6, C6, C2xC4, C23, C32, C32, C12, C12, A4, C2xC6, C2xC6, C22xC4, C3xC6, C3xC6, C2xC12, C2xA4, C22xC6, C33, C3xC12, C3xC12, C3xA4, C62, C62, C4xA4, C22xC12, C32xC6, C6xC12, C6xA4, C2xC62, C32xC12, C32xA4, C12xA4, C2xC6xC12, A4xC3xC6, A4xC3xC12
Quotients: C1, C2, C3, C4, C6, C32, C12, A4, C3xC6, C2xA4, C33, C3xC12, C3xA4, C4xA4, C32xC6, C6xA4, C32xC12, C32xA4, C12xA4, A4xC3xC6, A4xC3xC12

Smallest permutation representation of A4xC3xC12
On 108 points
Generators in S108
(1 76 40)(2 77 41)(3 78 42)(4 79 43)(5 80 44)(6 81 45)(7 82 46)(8 83 47)(9 84 48)(10 73 37)(11 74 38)(12 75 39)(13 52 95)(14 53 96)(15 54 85)(16 55 86)(17 56 87)(18 57 88)(19 58 89)(20 59 90)(21 60 91)(22 49 92)(23 50 93)(24 51 94)(25 98 71)(26 99 72)(27 100 61)(28 101 62)(29 102 63)(30 103 64)(31 104 65)(32 105 66)(33 106 67)(34 107 68)(35 108 69)(36 97 70)
(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)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(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)(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)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(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 69 91)(2 70 92)(3 71 93)(4 72 94)(5 61 95)(6 62 96)(7 63 85)(8 64 86)(9 65 87)(10 66 88)(11 67 89)(12 68 90)(13 80 27)(14 81 28)(15 82 29)(16 83 30)(17 84 31)(18 73 32)(19 74 33)(20 75 34)(21 76 35)(22 77 36)(23 78 25)(24 79 26)(37 105 57)(38 106 58)(39 107 59)(40 108 60)(41 97 49)(42 98 50)(43 99 51)(44 100 52)(45 101 53)(46 102 54)(47 103 55)(48 104 56)

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

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

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

144 conjugacy classes

class 1 2A2B2C3A···3H3I···3Z4A4B4C4D6A···6H6I···6X6Y···6AP12A···12P12Q···12AF12AG···12BP
order12223···33···344446···66···66···612···1212···1212···12
size11331···14···411331···13···34···41···13···34···4

144 irreducible representations

dim111111111333333
type++++
imageC1C2C3C3C4C6C6C12C12A4C2xA4C3xA4C4xA4C6xA4C12xA4
kernelA4xC3xC12A4xC3xC6C12xA4C2xC6xC12C32xA4C6xA4C2xC62C3xA4C62C3xC12C3xC6C12C32C6C3
# reps1124222424841182816

Matrix representation of A4xC3xC12 in GL4(F13) generated by

3000
0900
0090
0009
,
11000
0500
0050
0005
,
1000
0100
00120
010012
,
1000
01200
00120
0391
,
9000
0090
012108
0003
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] >;

A4xC3xC12 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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