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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 [×3], C3 [×4], C4, C4 [×3], C22 [×3], C6 [×4], C6 [×12], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C32, C12 [×16], C2×C6 [×12], C2×C8 [×3], M4(2) [×3], C4○D4, C3×C6, C3×C6 [×3], C3⋊C8 [×16], C2×C12 [×12], C3×D4 [×12], C3×Q8 [×4], C8○D4, C3×C12, C3×C12 [×3], C62 [×3], C2×C3⋊C8 [×12], C4.Dic3 [×12], C3×C4○D4 [×4], C324C8, C324C8 [×3], C6×C12 [×3], D4×C32 [×3], Q8×C32, D4.Dic3 [×4], C2×C324C8 [×3], C12.58D6 [×3], C32×C4○D4, D4.(C3⋊Dic3)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, Dic3 [×16], D6 [×12], C22×C4, C3⋊S3, C2×Dic3 [×24], C22×S3 [×4], C8○D4, C3⋊Dic3 [×4], C2×C3⋊S3 [×3], C22×Dic3 [×4], C2×C3⋊Dic3 [×6], C22×C3⋊S3, D4.Dic3 [×4], C22×C3⋊Dic3, D4.(C3⋊Dic3)

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

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

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

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

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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