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## G = A4⋊Dic9order 432 = 24·33

### The semidirect product of A4 and Dic9 acting via Dic9/C18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C9×A4 — A4⋊Dic9
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9×A4 — A4×C18 — A4⋊Dic9
 Lower central C9×A4 — 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=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 576 in 81 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, A4, C2×C6, C2×C6, C22⋊C4, C18, C18, C3×C6, C2×Dic3, C2×A4, C22×C6, C3×C9, Dic9, C3.A4, C2×C18, C2×C18, C3⋊Dic3, C3×A4, C6.D4, A4⋊C4, C3×C18, C2×Dic9, C2×C3.A4, C22×C18, C6×A4, C9⋊Dic3, C9×A4, C18.D4, C6.S4, C6.7S4, A4×C18, A4⋊Dic9
Quotients: C1, C2, C4, S3, Dic3, D9, C3⋊S3, S4, Dic9, C3⋊Dic3, A4⋊C4, C9⋊S3, C3⋊S4, C9⋊Dic3, C6.7S4, C9⋊S4, 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)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(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 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)
(1 45 25)(2 46 26)(3 47 27)(4 48 28)(5 49 29)(6 50 30)(7 51 31)(8 52 32)(9 53 33)(10 54 34)(11 37 35)(12 38 36)(13 39 19)(14 40 20)(15 41 21)(16 42 22)(17 43 23)(18 44 24)(55 83 97)(56 84 98)(57 85 99)(58 86 100)(59 87 101)(60 88 102)(61 89 103)(62 90 104)(63 73 105)(64 74 106)(65 75 107)(66 76 108)(67 77 91)(68 78 92)(69 79 93)(70 80 94)(71 81 95)(72 82 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)
(1 64 10 55)(2 63 11 72)(3 62 12 71)(4 61 13 70)(5 60 14 69)(6 59 15 68)(7 58 16 67)(8 57 17 66)(9 56 18 65)(19 80 28 89)(20 79 29 88)(21 78 30 87)(22 77 31 86)(23 76 32 85)(24 75 33 84)(25 74 34 83)(26 73 35 82)(27 90 36 81)(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)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(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), (1,10)(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), (1,45,25)(2,46,26)(3,47,27)(4,48,28)(5,49,29)(6,50,30)(7,51,31)(8,52,32)(9,53,33)(10,54,34)(11,37,35)(12,38,36)(13,39,19)(14,40,20)(15,41,21)(16,42,22)(17,43,23)(18,44,24)(55,83,97)(56,84,98)(57,85,99)(58,86,100)(59,87,101)(60,88,102)(61,89,103)(62,90,104)(63,73,105)(64,74,106)(65,75,107)(66,76,108)(67,77,91)(68,78,92)(69,79,93)(70,80,94)(71,81,95)(72,82,96), (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), (1,64,10,55)(2,63,11,72)(3,62,12,71)(4,61,13,70)(5,60,14,69)(6,59,15,68)(7,58,16,67)(8,57,17,66)(9,56,18,65)(19,80,28,89)(20,79,29,88)(21,78,30,87)(22,77,31,86)(23,76,32,85)(24,75,33,84)(25,74,34,83)(26,73,35,82)(27,90,36,81)(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)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(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), (1,10)(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), (1,45,25)(2,46,26)(3,47,27)(4,48,28)(5,49,29)(6,50,30)(7,51,31)(8,52,32)(9,53,33)(10,54,34)(11,37,35)(12,38,36)(13,39,19)(14,40,20)(15,41,21)(16,42,22)(17,43,23)(18,44,24)(55,83,97)(56,84,98)(57,85,99)(58,86,100)(59,87,101)(60,88,102)(61,89,103)(62,90,104)(63,73,105)(64,74,106)(65,75,107)(66,76,108)(67,77,91)(68,78,92)(69,79,93)(70,80,94)(71,81,95)(72,82,96), (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), (1,64,10,55)(2,63,11,72)(3,62,12,71)(4,61,13,70)(5,60,14,69)(6,59,15,68)(7,58,16,67)(8,57,17,66)(9,56,18,65)(19,80,28,89)(20,79,29,88)(21,78,30,87)(22,77,31,86)(23,76,32,85)(24,75,33,84)(25,74,34,83)(26,73,35,82)(27,90,36,81)(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),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(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)], [(1,10),(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)], [(1,45,25),(2,46,26),(3,47,27),(4,48,28),(5,49,29),(6,50,30),(7,51,31),(8,52,32),(9,53,33),(10,54,34),(11,37,35),(12,38,36),(13,39,19),(14,40,20),(15,41,21),(16,42,22),(17,43,23),(18,44,24),(55,83,97),(56,84,98),(57,85,99),(58,86,100),(59,87,101),(60,88,102),(61,89,103),(62,90,104),(63,73,105),(64,74,106),(65,75,107),(66,76,108),(67,77,91),(68,78,92),(69,79,93),(70,80,94),(71,81,95),(72,82,96)], [(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)], [(1,64,10,55),(2,63,11,72),(3,62,12,71),(4,61,13,70),(5,60,14,69),(6,59,15,68),(7,58,16,67),(8,57,17,66),(9,56,18,65),(19,80,28,89),(20,79,29,88),(21,78,30,87),(22,77,31,86),(23,76,32,85),(24,75,33,84),(25,74,34,83),(26,73,35,82),(27,90,36,81),(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)]])

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 9A 9B 9C 9D ··· 9I 18A 18B 18C 18D ··· 18I 18J ··· 18O order 1 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 6 6 9 9 9 9 ··· 9 18 18 18 18 ··· 18 18 ··· 18 size 1 1 3 3 2 8 8 8 54 54 54 54 2 6 6 8 8 8 2 2 2 8 ··· 8 2 2 2 6 ··· 6 8 ··· 8

42 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 3 3 6 6 6 6 type + + + + + - - - + - + + - + - image C1 C2 C4 S3 S3 S3 Dic3 Dic3 Dic3 D9 Dic9 S4 A4⋊C4 C3⋊S4 C6.7S4 C9⋊S4 A4⋊Dic9 kernel A4⋊Dic9 A4×C18 C9×A4 C2×C3.A4 C22×C18 C6×A4 C3.A4 C2×C18 C3×A4 C2×A4 A4 C18 C9 C6 C3 C2 C1 # reps 1 1 2 2 1 1 2 1 1 9 9 2 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 1 0 0 0 0 0 36 0 0 0 36 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 1 0 0 0 0 1 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 1 35 0 0 36 0 0 0 0 0 0 1
,
 26 17 0 0 0 20 6 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 6 0 0 0 6 0 0 0 0 0 0 36 1 35 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,36,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,1,1,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,36,0,0,0,1,0,0,0,0,35,0,1],[26,20,0,0,0,17,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,6,0,0,0,0,0,0,36,0,0,0,0,1,1,0,0,0,35,0,1] >;

A4⋊Dic9 in GAP, Magma, Sage, TeX

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

G:=Group("A4:Dic9");
// GroupNames label

G:=SmallGroup(432,254);
// by ID

G=gap.SmallGroup(432,254);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,14,170,1683,192,2524,9077,2287,5298,3989]);
// 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=e*a*e^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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