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## G = A4×Dic9order 432 = 24·33

### Direct product of A4 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — A4×Dic9
 Chief series C1 — C3 — C9 — C2×C18 — C22×C18 — A4×C18 — A4×Dic9
 Lower central C2×C18 — A4×Dic9
 Upper central C1 — C2

Generators and relations for A4×Dic9
G = < a,b,c,d,e | a2=b2=c3=d18=1, e2=d9, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 326 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×Dic3, C2×A4, C2×A4, C22×C6, C3×C9, Dic9, Dic9, C3.A4, C2×C18, C2×C18, C3×Dic3, C3×A4, C4×A4, C22×Dic3, C3×C18, C2×Dic9, C2×C3.A4, C22×C18, C6×A4, C3×Dic9, C9×A4, C22×Dic9, Dic3×A4, A4×C18, A4×Dic9
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, A4, D9, C3×S3, C2×A4, Dic9, C3×Dic3, C4×A4, C3×D9, S3×A4, C3×Dic9, Dic3×A4, A4×D9, A4×Dic9

Smallest permutation representation of A4×Dic9
On 108 points
Generators in S108
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)
(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)
(1 51 28)(2 52 29)(3 53 30)(4 54 31)(5 37 32)(6 38 33)(7 39 34)(8 40 35)(9 41 36)(10 42 19)(11 43 20)(12 44 21)(13 45 22)(14 46 23)(15 47 24)(16 48 25)(17 49 26)(18 50 27)(55 103 78)(56 104 79)(57 105 80)(58 106 81)(59 107 82)(60 108 83)(61 91 84)(62 92 85)(63 93 86)(64 94 87)(65 95 88)(66 96 89)(67 97 90)(68 98 73)(69 99 74)(70 100 75)(71 101 76)(72 102 77)
(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 70 10 61)(2 69 11 60)(3 68 12 59)(4 67 13 58)(5 66 14 57)(6 65 15 56)(7 64 16 55)(8 63 17 72)(9 62 18 71)(19 84 28 75)(20 83 29 74)(21 82 30 73)(22 81 31 90)(23 80 32 89)(24 79 33 88)(25 78 34 87)(26 77 35 86)(27 76 36 85)(37 96 46 105)(38 95 47 104)(39 94 48 103)(40 93 49 102)(41 92 50 101)(42 91 51 100)(43 108 52 99)(44 107 53 98)(45 106 54 97)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 9D ··· 9I 12A 12B 12C 12D 18A 18B 18C 18D ··· 18I 18J ··· 18O order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 9 9 9 9 ··· 9 12 12 12 12 18 18 18 18 ··· 18 18 ··· 18 size 1 1 3 3 2 4 4 8 8 9 9 27 27 2 4 4 6 6 8 8 2 2 2 8 ··· 8 36 36 36 36 2 2 2 6 ··· 6 8 ··· 8

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 6 6 6 6 type + + + - + - + + + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 D9 C3×S3 Dic9 C3×Dic3 C3×D9 C3×Dic9 A4 C2×A4 C4×A4 S3×A4 Dic3×A4 A4×D9 A4×Dic9 kernel A4×Dic9 A4×C18 C22×Dic9 C9×A4 C22×C18 C2×C18 C6×A4 C3×A4 C2×A4 C22×C6 A4 C2×C6 C23 C22 Dic9 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 3 2 3 2 6 6 1 1 2 1 1 3 3

Matrix representation of A4×Dic9 in GL5(𝔽37)

 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 1 13 0 0 0 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 13 0 0 0 36 0 0 0 0 0 36
,
 26 0 0 0 0 0 26 0 0 0 0 0 36 1 0 0 0 36 0 0 0 0 3 0 1
,
 11 31 0 0 0 6 17 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 23 30 0 0 0 7 14 0 0 0 0 0 31 0 0 0 0 0 31 0 0 0 0 0 31

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,13,36],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,13,0,36],[26,0,0,0,0,0,26,0,0,0,0,0,36,36,3,0,0,1,0,0,0,0,0,0,1],[11,6,0,0,0,31,17,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[23,7,0,0,0,30,14,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31] >;

A4×Dic9 in GAP, Magma, Sage, TeX

A_4\times {\rm Dic}_9
% in TeX

G:=Group("A4xDic9");
// GroupNames label

G:=SmallGroup(432,266);
// by ID

G=gap.SmallGroup(432,266);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-3,-3,42,514,221,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^18=1,e^2=d^9,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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