Copied to
clipboard

## G = D4×Dic9order 288 = 25·32

### Direct product of D4 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4×Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C22×Dic9 — D4×Dic9
 Lower central C9 — C18 — D4×Dic9
 Upper central C1 — C22 — C2×D4

Generators and relations for D4×Dic9
G = < a,b,c,d | a4=b2=c18=1, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 456 in 141 conjugacy classes, 70 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C9, Dic3 [×5], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C18 [×3], C18 [×4], C2×Dic3 [×8], C2×C12, C3×D4 [×4], C22×C6 [×2], C4×D4, Dic9 [×2], Dic9 [×3], C36 [×2], C2×C18, C2×C18 [×4], C2×C18 [×4], C4×Dic3, C4⋊Dic3, C6.D4 [×2], C22×Dic3 [×2], C6×D4, C2×Dic9 [×2], C2×Dic9 [×2], C2×Dic9 [×4], C2×C36, D4×C9 [×4], C22×C18 [×2], D4×Dic3, C4×Dic9, C4⋊Dic9, C18.D4 [×2], C22×Dic9 [×2], D4×C18, D4×Dic9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, Dic3 [×4], D6 [×3], C22×C4, C2×D4, C4○D4, D9, C2×Dic3 [×6], C22×S3, C4×D4, Dic9 [×4], D18 [×3], S3×D4, D42S3, C22×Dic3, C2×Dic9 [×6], C22×D9, D4×Dic3, D4×D9, D42D9, C22×Dic9, D4×Dic9

Smallest permutation representation of D4×Dic9
On 144 points
Generators in S144
(1 131 105 30)(2 132 106 31)(3 133 107 32)(4 134 108 33)(5 135 91 34)(6 136 92 35)(7 137 93 36)(8 138 94 19)(9 139 95 20)(10 140 96 21)(11 141 97 22)(12 142 98 23)(13 143 99 24)(14 144 100 25)(15 127 101 26)(16 128 102 27)(17 129 103 28)(18 130 104 29)(37 84 117 58)(38 85 118 59)(39 86 119 60)(40 87 120 61)(41 88 121 62)(42 89 122 63)(43 90 123 64)(44 73 124 65)(45 74 125 66)(46 75 126 67)(47 76 109 68)(48 77 110 69)(49 78 111 70)(50 79 112 71)(51 80 113 72)(52 81 114 55)(53 82 115 56)(54 83 116 57)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 19)(18 20)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 73)(54 74)(55 123)(56 124)(57 125)(58 126)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)(91 144)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)(101 136)(102 137)(103 138)(104 139)(105 140)(106 141)(107 142)(108 143)
(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 56 10 65)(2 55 11 64)(3 72 12 63)(4 71 13 62)(5 70 14 61)(6 69 15 60)(7 68 16 59)(8 67 17 58)(9 66 18 57)(19 126 28 117)(20 125 29 116)(21 124 30 115)(22 123 31 114)(23 122 32 113)(24 121 33 112)(25 120 34 111)(26 119 35 110)(27 118 36 109)(37 138 46 129)(38 137 47 128)(39 136 48 127)(40 135 49 144)(41 134 50 143)(42 133 51 142)(43 132 52 141)(44 131 53 140)(45 130 54 139)(73 105 82 96)(74 104 83 95)(75 103 84 94)(76 102 85 93)(77 101 86 92)(78 100 87 91)(79 99 88 108)(80 98 89 107)(81 97 90 106)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G ··· 4L 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 12A 12B 18A ··· 18I 18J ··· 18U 36A ··· 36F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 6 6 9 9 9 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 2 2 2 2 9 9 9 9 18 ··· 18 2 2 2 4 4 4 4 2 2 2 4 4 2 ··· 2 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + - + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 Dic3 D6 C4○D4 D9 D18 Dic9 D18 S3×D4 D4⋊2S3 D4×D9 D4⋊2D9 kernel D4×Dic9 C4×Dic9 C4⋊Dic9 C18.D4 C22×Dic9 D4×C18 D4×C9 C6×D4 Dic9 C2×C12 C3×D4 C22×C6 C18 C2×D4 C2×C4 D4 C23 C6 C6 C2 C2 # reps 1 1 1 2 2 1 8 1 2 1 4 2 2 3 3 12 6 1 1 3 3

Matrix representation of D4×Dic9 in GL5(𝔽37)

 36 0 0 0 0 0 1 35 0 0 0 1 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 36 0 0 0 0 0 1 35 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 36 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 20 0 0 0 17 31
,
 6 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 36 1

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,1,0,0,0,35,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,1,0,0,0,0,35,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,17,0,0,0,20,31],[6,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,36,0,0,0,0,1] >;

D4×Dic9 in GAP, Magma, Sage, TeX

D_4\times {\rm Dic}_9
% in TeX

G:=Group("D4xDic9");
// GroupNames label

G:=SmallGroup(288,144);
// by ID

G=gap.SmallGroup(288,144);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,219,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^18=1,d^2=c^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽