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## G = A4×C62order 432 = 24·33

### Direct product of C62 and A4

Aliases: A4×C62, C242C33, (C2×C6)⋊3C62, (C2×C62)⋊9C6, C22⋊(C3×C62), C23⋊(C32×C6), C6216(C2×C6), (C22×C62)⋊4C3, (C23×C6)⋊2C32, (C22×C6)⋊2(C3×C6), SmallGroup(432,770)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C62
 Chief series C1 — C22 — C2×C6 — C62 — C32×A4 — A4×C3×C6 — A4×C62
 Lower central C22 — A4×C62
 Upper central C1 — C62

Generators and relations for A4×C62
G = < a,b,c,d,e | a6=b6=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: 952 in 394 conjugacy classes, 170 normal (10 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C23, C32, C32, A4, C2×C6, C2×C6, C24, C3×C6, C3×C6, C2×A4, C22×C6, C22×C6, C33, C3×A4, C62, C62, C22×A4, C23×C6, C32×C6, C6×A4, C2×C62, C2×C62, C32×A4, C3×C62, A4×C2×C6, C22×C62, A4×C3×C6, A4×C62
Quotients: C1, C2, C3, C22, C6, C32, A4, C2×C6, C3×C6, C2×A4, C33, C3×A4, C62, C22×A4, C32×C6, C6×A4, C32×A4, C3×C62, A4×C2×C6, A4×C3×C6, A4×C62

Smallest permutation representation of A4×C62
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)
(1 81 23 74 27 69)(2 82 24 75 28 70)(3 83 19 76 29 71)(4 84 20 77 30 72)(5 79 21 78 25 67)(6 80 22 73 26 68)(7 65 107 58 18 53)(8 66 108 59 13 54)(9 61 103 60 14 49)(10 62 104 55 15 50)(11 63 105 56 16 51)(12 64 106 57 17 52)(31 96 43 85 42 97)(32 91 44 86 37 98)(33 92 45 87 38 99)(34 93 46 88 39 100)(35 94 47 89 40 101)(36 95 48 90 41 102)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)(55 58)(56 59)(57 60)(61 64)(62 65)(63 66)(85 88)(86 89)(87 90)(91 94)(92 95)(93 96)(97 100)(98 101)(99 102)(103 106)(104 107)(105 108)
(1 4)(2 5)(3 6)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(67 70)(68 71)(69 72)(73 76)(74 77)(75 78)(79 82)(80 83)(81 84)(85 88)(86 89)(87 90)(91 94)(92 95)(93 96)(97 100)(98 101)(99 102)
(1 65 31)(2 66 32)(3 61 33)(4 62 34)(5 63 35)(6 64 36)(7 97 69)(8 98 70)(9 99 71)(10 100 72)(11 101 67)(12 102 68)(13 86 75)(14 87 76)(15 88 77)(16 89 78)(17 90 73)(18 85 74)(19 60 45)(20 55 46)(21 56 47)(22 57 48)(23 58 43)(24 59 44)(25 51 40)(26 52 41)(27 53 42)(28 54 37)(29 49 38)(30 50 39)(79 105 94)(80 106 95)(81 107 96)(82 108 91)(83 103 92)(84 104 93)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 3I ··· 3Z 6A ··· 6X 6Y ··· 6BD 6BE ··· 6DF order 1 2 2 2 2 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 3 3 3 3 1 ··· 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

144 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 A4 C2×A4 C3×A4 C6×A4 kernel A4×C62 A4×C3×C6 A4×C2×C6 C22×C62 C6×A4 C2×C62 C62 C3×C6 C2×C6 C6 # reps 1 3 24 2 72 6 1 3 8 24

Matrix representation of A4×C62 in GL5(𝔽7)

 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5
,
 1 0 0 0 0 0 3 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 1
,
 4 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 4 0 0 4 0 0

G:=sub<GL(5,GF(7))| [5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,3,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,1],[4,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,4,0] >;

A4×C62 in GAP, Magma, Sage, TeX

A_4\times C_6^2
% in TeX

G:=Group("A4xC6^2");
// GroupNames label

G:=SmallGroup(432,770);
// by ID

G=gap.SmallGroup(432,770);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,2287,3989]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^6=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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