Copied to
clipboard

## G = Dic9.D4order 288 = 25·32

### 1st non-split extension by Dic9 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — Dic9.D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — D18⋊C4 — Dic9.D4
 Lower central C9 — C2×C18 — Dic9.D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic9.D4
G = < a,b,c,d | a18=c4=1, b2=d2=a9, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a9b, dcd-1=a9c-1 >

Subgroups: 524 in 114 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, D9, C18, C18, Dic6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4.4D4, Dic9, Dic9, C36, D18, C2×C18, C2×C18, C4×Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, C2×C3⋊D4, Dic18, C2×Dic9, C9⋊D4, C2×C36, C22×D9, C22×C18, C23.11D6, C4×Dic9, D18⋊C4, C18.D4, C9×C22⋊C4, C2×Dic18, C2×C9⋊D4, Dic9.D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D9, C22×S3, C4.4D4, D18, C4○D12, S3×D4, D42S3, C22×D9, C23.11D6, D365C2, D4×D9, D42D9, Dic9.D4

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

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

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

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

54 conjugacy classes

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

54 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 C2 S3 D4 D6 D6 C4○D4 D9 D18 D18 C4○D12 D36⋊5C2 S3×D4 D4⋊2S3 D4×D9 D4⋊2D9 kernel Dic9.D4 C4×Dic9 D18⋊C4 C18.D4 C9×C22⋊C4 C2×Dic18 C2×C9⋊D4 C3×C22⋊C4 Dic9 C2×C12 C22×C6 C18 C22⋊C4 C2×C4 C23 C6 C2 C6 C6 C2 C2 # reps 1 1 2 1 1 1 1 1 2 2 1 4 3 6 3 4 12 1 1 3 3

Matrix representation of Dic9.D4 in GL6(𝔽37)

 17 11 0 0 0 0 26 6 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 36 36 0 0 0 0 0 0 0 36 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 35 36
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 6 0 0 0 0 31 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 36 0 0 0 0 0 1 1 0 0 0 0 0 0 0 6 0 0 0 0 6 0 0 0 0 0 0 0 6 0 0 0 0 0 25 31

G:=sub<GL(6,GF(37))| [17,26,0,0,0,0,11,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,36,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,36,0,0,0,0,0,0,0,1,35,0,0,0,0,1,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,31,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[36,1,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,6,0,0,0,0,0,0,0,6,25,0,0,0,0,0,31] >;

Dic9.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_9.D_4
% in TeX

G:=Group("Dic9.D4");
// GroupNames label

G:=SmallGroup(288,95);
// by ID

G=gap.SmallGroup(288,95);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,219,6725,292,9414]);
// Polycyclic

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

׿
×
𝔽