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## G = D4.(C3⋊Dic3)  order 288 = 25·32

### The non-split extension by D4 of C3⋊Dic3 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D4.(C3⋊Dic3)
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C2×C32⋊4C8 — D4.(C3⋊Dic3)
 Lower central C32 — C3×C6 — D4.(C3⋊Dic3)
 Upper central C1 — C4 — C4○D4

Generators and relations for D4.(C3⋊Dic3)
G = < a,b,c,d,e | a4=b2=c3=1, d6=a2, e2=a2d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >

Subgroups: 380 in 186 conjugacy classes, 113 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, C32, C12, C2×C6, C2×C8, M4(2), C4○D4, C3×C6, C3×C6, C3⋊C8, C2×C12, C3×D4, C3×Q8, C8○D4, C3×C12, C3×C12, C62, C2×C3⋊C8, C4.Dic3, C3×C4○D4, C324C8, C324C8, C6×C12, D4×C32, Q8×C32, D4.Dic3, C2×C324C8, C12.58D6, C32×C4○D4, D4.(C3⋊Dic3)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C3⋊S3, C2×Dic3, C22×S3, C8○D4, C3⋊Dic3, C2×C3⋊S3, C22×Dic3, C2×C3⋊Dic3, C22×C3⋊S3, D4.Dic3, C22×C3⋊Dic3, D4.(C3⋊Dic3)

Smallest permutation representation of D4.(C3⋊Dic3)
On 144 points
Generators in S144
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 51 19 57)(14 52 20 58)(15 53 21 59)(16 54 22 60)(17 55 23 49)(18 56 24 50)(37 75 43 81)(38 76 44 82)(39 77 45 83)(40 78 46 84)(41 79 47 73)(42 80 48 74)(61 121 67 127)(62 122 68 128)(63 123 69 129)(64 124 70 130)(65 125 71 131)(66 126 72 132)(85 114 91 120)(86 115 92 109)(87 116 93 110)(88 117 94 111)(89 118 95 112)(90 119 96 113)(97 136 103 142)(98 137 104 143)(99 138 105 144)(100 139 106 133)(101 140 107 134)(102 141 108 135)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)
(1 75 118)(2 76 119)(3 77 120)(4 78 109)(5 79 110)(6 80 111)(7 81 112)(8 82 113)(9 83 114)(10 84 115)(11 73 116)(12 74 117)(13 107 122)(14 108 123)(15 97 124)(16 98 125)(17 99 126)(18 100 127)(19 101 128)(20 102 129)(21 103 130)(22 104 131)(23 105 132)(24 106 121)(25 37 89)(26 38 90)(27 39 91)(28 40 92)(29 41 93)(30 42 94)(31 43 95)(32 44 96)(33 45 85)(34 46 86)(35 47 87)(36 48 88)(49 144 66)(50 133 67)(51 134 68)(52 135 69)(53 136 70)(54 137 71)(55 138 72)(56 139 61)(57 140 62)(58 141 63)(59 142 64)(60 143 65)
(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)
(1 129 10 126 7 123 4 132)(2 122 11 131 8 128 5 125)(3 127 12 124 9 121 6 130)(13 116 22 113 19 110 16 119)(14 109 23 118 20 115 17 112)(15 114 24 111 21 120 18 117)(25 69 34 66 31 63 28 72)(26 62 35 71 32 68 29 65)(27 67 36 64 33 61 30 70)(37 135 46 144 43 141 40 138)(38 140 47 137 44 134 41 143)(39 133 48 142 45 139 42 136)(49 95 58 92 55 89 52 86)(50 88 59 85 56 94 53 91)(51 93 60 90 57 87 54 96)(73 104 82 101 79 98 76 107)(74 97 83 106 80 103 77 100)(75 102 84 99 81 108 78 105)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E ··· 6P 8A 8B 8C 8D 8E ··· 8J 12A ··· 12H 12I ··· 12T order 1 2 2 2 2 3 3 3 3 4 4 4 4 4 6 6 6 6 6 ··· 6 8 8 8 8 8 ··· 8 12 ··· 12 12 ··· 12 size 1 1 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 4 ··· 4 9 9 9 9 18 ··· 18 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 C4 S3 D6 Dic3 Dic3 C8○D4 D4.Dic3 kernel D4.(C3⋊Dic3) C2×C32⋊4C8 C12.58D6 C32×C4○D4 D4×C32 Q8×C32 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C32 C3 # reps 1 3 3 1 6 2 4 12 12 4 4 8

Matrix representation of D4.(C3⋊Dic3) in GL6(𝔽73)

 72 2 0 0 0 0 72 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 51 0 0 0 0 0 0 51 0 0 0 0 0 0 60 2 0 0 0 0 62 13 0 0 0 0 0 0 19 29 0 0 0 0 48 54

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,0,0,0,0,0,0,51,0,0,0,0,0,0,60,62,0,0,0,0,2,13,0,0,0,0,0,0,19,48,0,0,0,0,29,54] >;

D4.(C3⋊Dic3) in GAP, Magma, Sage, TeX

D_4.(C_3\rtimes {\rm Dic}_3)
% in TeX

G:=Group("D4.(C3:Dic3)");
// GroupNames label

G:=SmallGroup(288,805);
// by ID

G=gap.SmallGroup(288,805);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,219,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^3=1,d^6=a^2,e^2=a^2*d^3,b*a*b=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,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations

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