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## G = Dic9⋊A4order 432 = 24·33

### The semidirect product of Dic9 and A4 acting via A4/C22=C3

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic9⋊A4
G = < a,b,c,d,e | a18=c2=d2=e3=1, b2=a9, bab-1=a-1, ac=ca, ad=da, eae-1=a13, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 326 in 61 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, A4, C2×C6, C2×C6, C22×C4, C18, C18, C3×C6, C2×Dic3, C2×A4, C22×C6, 3- 1+2, Dic9, Dic9, C3.A4, C2×C18, C2×C18, C3×Dic3, C3×A4, C4×A4, C22×Dic3, C2×3- 1+2, C2×Dic9, C2×C3.A4, C22×C18, C6×A4, C9⋊C12, C9⋊A4, C22×Dic9, Dic3×A4, C2×C9⋊A4, Dic9⋊A4
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, A4, C3×S3, C2×A4, C3×Dic3, C4×A4, C9⋊C6, S3×A4, C9⋊C12, Dic3×A4, D9⋊A4, Dic9⋊A4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 9A 9B 9C 12A 12B 12C 12D 18A ··· 18G 18H 18I order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 9 9 9 12 12 12 12 18 ··· 18 18 18 size 1 1 3 3 2 12 12 9 9 27 27 2 6 6 12 12 6 24 24 36 36 36 36 6 ··· 6 24 24

32 irreducible representations

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

Matrix representation of Dic9⋊A4 in GL6(𝔽37)

 31 26 0 0 0 0 11 20 0 0 0 0 0 1 11 31 0 0 14 15 6 17 0 0 29 2 0 0 6 17 27 0 0 0 20 26
,
 36 9 0 0 0 0 8 1 0 0 0 0 27 0 9 1 0 0 6 16 29 28 0 0 11 0 0 0 9 1 23 12 0 0 29 28
,
 36 0 0 0 0 0 0 36 0 0 0 0 18 18 1 0 0 0 18 18 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 1 0 0 0 0 19 19 36 0 0 0 19 19 0 36 0 0 5 5 0 0 36 0 5 5 0 0 0 36
,
 0 0 36 1 0 0 10 10 35 36 0 0 0 0 27 0 1 0 0 0 27 0 0 1 0 0 11 0 0 0 1 0 11 0 0 0

G:=sub<GL(6,GF(37))| [31,11,0,14,29,27,26,20,1,15,2,0,0,0,11,6,0,0,0,0,31,17,0,0,0,0,0,0,6,20,0,0,0,0,17,26],[36,8,27,6,11,23,9,1,0,16,0,12,0,0,9,29,0,0,0,0,1,28,0,0,0,0,0,0,9,29,0,0,0,0,1,28],[36,0,18,18,0,0,0,36,18,18,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,19,19,5,5,0,1,19,19,5,5,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[0,10,0,0,0,1,0,10,0,0,0,0,36,35,27,27,11,11,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

Dic9⋊A4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(432,265);
// by ID

G=gap.SmallGroup(432,265);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-3,-3,42,514,221,10085,4044,292,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^18=c^2=d^2=e^3=1,b^2=a^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^13,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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