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## G = Dic9.2A4order 432 = 24·33

### The non-split extension by Dic9 of A4 acting through Inn(Dic9)

Aliases: Dic9.2A4, SL2(𝔽3)⋊2D9, C6.2(S3×A4), C2.2(A4×D9), C93(C4.A4), C18.7(C2×A4), (Q8×C9).3C6, Q8.2(C3×D9), Q83D93C3, (C9×SL2(𝔽3))⋊2C2, C3.2(Dic3.A4), (C3×SL2(𝔽3)).6S3, (C3×Q8).12(C3×S3), SmallGroup(432,262)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C9 — Dic9.2A4
 Chief series C1 — C3 — C6 — C18 — Q8×C9 — C9×SL2(𝔽3) — Dic9.2A4
 Lower central Q8×C9 — Dic9.2A4
 Upper central C1 — C2

Generators and relations for Dic9.2A4
G = < a,b,c,d,e | a18=e3=1, b2=c2=d2=a9, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a9c, ece-1=a9cd, ede-1=c >

Smallest permutation representation of Dic9.2A4
On 144 points
Generators in S144
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 88 10 79)(2 87 11 78)(3 86 12 77)(4 85 13 76)(5 84 14 75)(6 83 15 74)(7 82 16 73)(8 81 17 90)(9 80 18 89)(19 107 28 98)(20 106 29 97)(21 105 30 96)(22 104 31 95)(23 103 32 94)(24 102 33 93)(25 101 34 92)(26 100 35 91)(27 99 36 108)(37 122 46 113)(38 121 47 112)(39 120 48 111)(40 119 49 110)(41 118 50 109)(42 117 51 126)(43 116 52 125)(44 115 53 124)(45 114 54 123)(55 144 64 135)(56 143 65 134)(57 142 66 133)(58 141 67 132)(59 140 68 131)(60 139 69 130)(61 138 70 129)(62 137 71 128)(63 136 72 127)
(1 35 10 26)(2 36 11 27)(3 19 12 28)(4 20 13 29)(5 21 14 30)(6 22 15 31)(7 23 16 32)(8 24 17 33)(9 25 18 34)(37 62 46 71)(38 63 47 72)(39 64 48 55)(40 65 49 56)(41 66 50 57)(42 67 51 58)(43 68 52 59)(44 69 53 60)(45 70 54 61)(73 94 82 103)(74 95 83 104)(75 96 84 105)(76 97 85 106)(77 98 86 107)(78 99 87 108)(79 100 88 91)(80 101 89 92)(81 102 90 93)(109 142 118 133)(110 143 119 134)(111 144 120 135)(112 127 121 136)(113 128 122 137)(114 129 123 138)(115 130 124 139)(116 131 125 140)(117 132 126 141)
(1 50 10 41)(2 51 11 42)(3 52 12 43)(4 53 13 44)(5 54 14 45)(6 37 15 46)(7 38 16 47)(8 39 17 48)(9 40 18 49)(19 68 28 59)(20 69 29 60)(21 70 30 61)(22 71 31 62)(23 72 32 63)(24 55 33 64)(25 56 34 65)(26 57 35 66)(27 58 36 67)(73 112 82 121)(74 113 83 122)(75 114 84 123)(76 115 85 124)(77 116 86 125)(78 117 87 126)(79 118 88 109)(80 119 89 110)(81 120 90 111)(91 133 100 142)(92 134 101 143)(93 135 102 144)(94 136 103 127)(95 137 104 128)(96 138 105 129)(97 139 106 130)(98 140 107 131)(99 141 108 132)
(19 52 59)(20 53 60)(21 54 61)(22 37 62)(23 38 63)(24 39 64)(25 40 65)(26 41 66)(27 42 67)(28 43 68)(29 44 69)(30 45 70)(31 46 71)(32 47 72)(33 48 55)(34 49 56)(35 50 57)(36 51 58)(91 109 142)(92 110 143)(93 111 144)(94 112 127)(95 113 128)(96 114 129)(97 115 130)(98 116 131)(99 117 132)(100 118 133)(101 119 134)(102 120 135)(103 121 136)(104 122 137)(105 123 138)(106 124 139)(107 125 140)(108 126 141)

G:=sub<Sym(144)| (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,88,10,79)(2,87,11,78)(3,86,12,77)(4,85,13,76)(5,84,14,75)(6,83,15,74)(7,82,16,73)(8,81,17,90)(9,80,18,89)(19,107,28,98)(20,106,29,97)(21,105,30,96)(22,104,31,95)(23,103,32,94)(24,102,33,93)(25,101,34,92)(26,100,35,91)(27,99,36,108)(37,122,46,113)(38,121,47,112)(39,120,48,111)(40,119,49,110)(41,118,50,109)(42,117,51,126)(43,116,52,125)(44,115,53,124)(45,114,54,123)(55,144,64,135)(56,143,65,134)(57,142,66,133)(58,141,67,132)(59,140,68,131)(60,139,69,130)(61,138,70,129)(62,137,71,128)(63,136,72,127), (1,35,10,26)(2,36,11,27)(3,19,12,28)(4,20,13,29)(5,21,14,30)(6,22,15,31)(7,23,16,32)(8,24,17,33)(9,25,18,34)(37,62,46,71)(38,63,47,72)(39,64,48,55)(40,65,49,56)(41,66,50,57)(42,67,51,58)(43,68,52,59)(44,69,53,60)(45,70,54,61)(73,94,82,103)(74,95,83,104)(75,96,84,105)(76,97,85,106)(77,98,86,107)(78,99,87,108)(79,100,88,91)(80,101,89,92)(81,102,90,93)(109,142,118,133)(110,143,119,134)(111,144,120,135)(112,127,121,136)(113,128,122,137)(114,129,123,138)(115,130,124,139)(116,131,125,140)(117,132,126,141), (1,50,10,41)(2,51,11,42)(3,52,12,43)(4,53,13,44)(5,54,14,45)(6,37,15,46)(7,38,16,47)(8,39,17,48)(9,40,18,49)(19,68,28,59)(20,69,29,60)(21,70,30,61)(22,71,31,62)(23,72,32,63)(24,55,33,64)(25,56,34,65)(26,57,35,66)(27,58,36,67)(73,112,82,121)(74,113,83,122)(75,114,84,123)(76,115,85,124)(77,116,86,125)(78,117,87,126)(79,118,88,109)(80,119,89,110)(81,120,90,111)(91,133,100,142)(92,134,101,143)(93,135,102,144)(94,136,103,127)(95,137,104,128)(96,138,105,129)(97,139,106,130)(98,140,107,131)(99,141,108,132), (19,52,59)(20,53,60)(21,54,61)(22,37,62)(23,38,63)(24,39,64)(25,40,65)(26,41,66)(27,42,67)(28,43,68)(29,44,69)(30,45,70)(31,46,71)(32,47,72)(33,48,55)(34,49,56)(35,50,57)(36,51,58)(91,109,142)(92,110,143)(93,111,144)(94,112,127)(95,113,128)(96,114,129)(97,115,130)(98,116,131)(99,117,132)(100,118,133)(101,119,134)(102,120,135)(103,121,136)(104,122,137)(105,123,138)(106,124,139)(107,125,140)(108,126,141)>;

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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,88,10,79)(2,87,11,78)(3,86,12,77)(4,85,13,76)(5,84,14,75)(6,83,15,74)(7,82,16,73)(8,81,17,90)(9,80,18,89)(19,107,28,98)(20,106,29,97)(21,105,30,96)(22,104,31,95)(23,103,32,94)(24,102,33,93)(25,101,34,92)(26,100,35,91)(27,99,36,108)(37,122,46,113)(38,121,47,112)(39,120,48,111)(40,119,49,110)(41,118,50,109)(42,117,51,126)(43,116,52,125)(44,115,53,124)(45,114,54,123)(55,144,64,135)(56,143,65,134)(57,142,66,133)(58,141,67,132)(59,140,68,131)(60,139,69,130)(61,138,70,129)(62,137,71,128)(63,136,72,127), (1,35,10,26)(2,36,11,27)(3,19,12,28)(4,20,13,29)(5,21,14,30)(6,22,15,31)(7,23,16,32)(8,24,17,33)(9,25,18,34)(37,62,46,71)(38,63,47,72)(39,64,48,55)(40,65,49,56)(41,66,50,57)(42,67,51,58)(43,68,52,59)(44,69,53,60)(45,70,54,61)(73,94,82,103)(74,95,83,104)(75,96,84,105)(76,97,85,106)(77,98,86,107)(78,99,87,108)(79,100,88,91)(80,101,89,92)(81,102,90,93)(109,142,118,133)(110,143,119,134)(111,144,120,135)(112,127,121,136)(113,128,122,137)(114,129,123,138)(115,130,124,139)(116,131,125,140)(117,132,126,141), (1,50,10,41)(2,51,11,42)(3,52,12,43)(4,53,13,44)(5,54,14,45)(6,37,15,46)(7,38,16,47)(8,39,17,48)(9,40,18,49)(19,68,28,59)(20,69,29,60)(21,70,30,61)(22,71,31,62)(23,72,32,63)(24,55,33,64)(25,56,34,65)(26,57,35,66)(27,58,36,67)(73,112,82,121)(74,113,83,122)(75,114,84,123)(76,115,85,124)(77,116,86,125)(78,117,87,126)(79,118,88,109)(80,119,89,110)(81,120,90,111)(91,133,100,142)(92,134,101,143)(93,135,102,144)(94,136,103,127)(95,137,104,128)(96,138,105,129)(97,139,106,130)(98,140,107,131)(99,141,108,132), (19,52,59)(20,53,60)(21,54,61)(22,37,62)(23,38,63)(24,39,64)(25,40,65)(26,41,66)(27,42,67)(28,43,68)(29,44,69)(30,45,70)(31,46,71)(32,47,72)(33,48,55)(34,49,56)(35,50,57)(36,51,58)(91,109,142)(92,110,143)(93,111,144)(94,112,127)(95,113,128)(96,114,129)(97,115,130)(98,116,131)(99,117,132)(100,118,133)(101,119,134)(102,120,135)(103,121,136)(104,122,137)(105,123,138)(106,124,139)(107,125,140)(108,126,141) );

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),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,88,10,79),(2,87,11,78),(3,86,12,77),(4,85,13,76),(5,84,14,75),(6,83,15,74),(7,82,16,73),(8,81,17,90),(9,80,18,89),(19,107,28,98),(20,106,29,97),(21,105,30,96),(22,104,31,95),(23,103,32,94),(24,102,33,93),(25,101,34,92),(26,100,35,91),(27,99,36,108),(37,122,46,113),(38,121,47,112),(39,120,48,111),(40,119,49,110),(41,118,50,109),(42,117,51,126),(43,116,52,125),(44,115,53,124),(45,114,54,123),(55,144,64,135),(56,143,65,134),(57,142,66,133),(58,141,67,132),(59,140,68,131),(60,139,69,130),(61,138,70,129),(62,137,71,128),(63,136,72,127)], [(1,35,10,26),(2,36,11,27),(3,19,12,28),(4,20,13,29),(5,21,14,30),(6,22,15,31),(7,23,16,32),(8,24,17,33),(9,25,18,34),(37,62,46,71),(38,63,47,72),(39,64,48,55),(40,65,49,56),(41,66,50,57),(42,67,51,58),(43,68,52,59),(44,69,53,60),(45,70,54,61),(73,94,82,103),(74,95,83,104),(75,96,84,105),(76,97,85,106),(77,98,86,107),(78,99,87,108),(79,100,88,91),(80,101,89,92),(81,102,90,93),(109,142,118,133),(110,143,119,134),(111,144,120,135),(112,127,121,136),(113,128,122,137),(114,129,123,138),(115,130,124,139),(116,131,125,140),(117,132,126,141)], [(1,50,10,41),(2,51,11,42),(3,52,12,43),(4,53,13,44),(5,54,14,45),(6,37,15,46),(7,38,16,47),(8,39,17,48),(9,40,18,49),(19,68,28,59),(20,69,29,60),(21,70,30,61),(22,71,31,62),(23,72,32,63),(24,55,33,64),(25,56,34,65),(26,57,35,66),(27,58,36,67),(73,112,82,121),(74,113,83,122),(75,114,84,123),(76,115,85,124),(77,116,86,125),(78,117,87,126),(79,118,88,109),(80,119,89,110),(81,120,90,111),(91,133,100,142),(92,134,101,143),(93,135,102,144),(94,136,103,127),(95,137,104,128),(96,138,105,129),(97,139,106,130),(98,140,107,131),(99,141,108,132)], [(19,52,59),(20,53,60),(21,54,61),(22,37,62),(23,38,63),(24,39,64),(25,40,65),(26,41,66),(27,42,67),(28,43,68),(29,44,69),(30,45,70),(31,46,71),(32,47,72),(33,48,55),(34,49,56),(35,50,57),(36,51,58),(91,109,142),(92,110,143),(93,111,144),(94,112,127),(95,113,128),(96,114,129),(97,115,130),(98,116,131),(99,117,132),(100,118,133),(101,119,134),(102,120,135),(103,121,136),(104,122,137),(105,123,138),(106,124,139),(107,125,140),(108,126,141)]])

42 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 9A 9B 9C 9D ··· 9I 12A 12B 12C 12D 12E 18A 18B 18C 18D ··· 18I 36A 36B 36C order 1 2 2 3 3 3 3 3 4 4 4 6 6 6 6 6 9 9 9 9 ··· 9 12 12 12 12 12 18 18 18 18 ··· 18 36 36 36 size 1 1 54 2 4 4 8 8 6 9 9 2 4 4 8 8 2 2 2 8 ··· 8 12 36 36 36 36 2 2 2 8 ··· 8 12 12 12

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 3 3 4 4 4 4 6 6 type + + + + + + + + + + image C1 C2 C3 C6 S3 D9 C3×S3 C4.A4 C3×D9 A4 C2×A4 Dic3.A4 Dic3.A4 Dic9.2A4 Dic9.2A4 S3×A4 A4×D9 kernel Dic9.2A4 C9×SL2(𝔽3) Q8⋊3D9 Q8×C9 C3×SL2(𝔽3) SL2(𝔽3) C3×Q8 C9 Q8 Dic9 C18 C3 C3 C1 C1 C6 C2 # reps 1 1 2 2 1 3 2 6 6 1 1 1 2 3 6 1 3

Matrix representation of Dic9.2A4 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 33 0 0 0 3 9
,
 6 0 0 0 0 6 0 0 0 0 5 34 0 0 8 32
,
 0 1 0 0 36 0 0 0 0 0 1 0 0 0 0 1
,
 11 10 0 0 10 26 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 10 26 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,33,3,0,0,0,9],[6,0,0,0,0,6,0,0,0,0,5,8,0,0,34,32],[0,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[11,10,0,0,10,26,0,0,0,0,1,0,0,0,0,1],[1,10,0,0,0,26,0,0,0,0,1,0,0,0,0,1] >;

Dic9.2A4 in GAP, Magma, Sage, TeX

{\rm Dic}_9._2A_4
% in TeX

G:=Group("Dic9.2A4");
// GroupNames label

G:=SmallGroup(432,262);
// by ID

G=gap.SmallGroup(432,262);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-3,-2,-3,504,198,268,94,409,192,6724,452,14118]);
// Polycyclic

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

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