Copied to
clipboard

## G = D4.10D18order 288 = 25·32

### The non-split extension by D4 of D18 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4.10D18
 Chief series C1 — C3 — C9 — C18 — D18 — C4×D9 — Q8×D9 — D4.10D18
 Lower central C9 — C18 — D4.10D18
 Upper central C1 — C2 — C4○D4

Generators and relations for D4.10D18
G = < a,b,c,d | a4=b2=1, c18=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c17 >

Subgroups: 784 in 219 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3 [×2], C6, C6 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], C9, Dic3 [×6], C12, C12 [×3], D6 [×2], C2×C6 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×9], D9 [×2], C18, C18 [×3], Dic6 [×9], C4×S3 [×6], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, 2- 1+4, Dic9 [×6], C36, C36 [×3], D18 [×2], C2×C18 [×3], C2×Dic6 [×3], C4○D12 [×3], D42S3 [×6], S3×Q8 [×2], C3×C4○D4, Dic18 [×9], C4×D9 [×6], D36, C2×Dic9 [×6], C9⋊D4 [×6], C2×C36 [×3], D4×C9 [×3], Q8×C9, Q8○D12, C2×Dic18 [×3], D365C2 [×3], D42D9 [×6], Q8×D9 [×2], C9×C4○D4, D4.10D18
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, D9, C22×S3 [×7], 2- 1+4, D18 [×7], S3×C23, C22×D9 [×7], Q8○D12, C23×D9, D4.10D18

Smallest permutation representation of D4.10D18
On 144 points
Generators in S144
```(1 72 19 54)(2 55 20 37)(3 38 21 56)(4 57 22 39)(5 40 23 58)(6 59 24 41)(7 42 25 60)(8 61 26 43)(9 44 27 62)(10 63 28 45)(11 46 29 64)(12 65 30 47)(13 48 31 66)(14 67 32 49)(15 50 33 68)(16 69 34 51)(17 52 35 70)(18 71 36 53)(73 112 91 130)(74 131 92 113)(75 114 93 132)(76 133 94 115)(77 116 95 134)(78 135 96 117)(79 118 97 136)(80 137 98 119)(81 120 99 138)(82 139 100 121)(83 122 101 140)(84 141 102 123)(85 124 103 142)(86 143 104 125)(87 126 105 144)(88 109 106 127)(89 128 107 110)(90 111 108 129)
(1 117)(2 136)(3 119)(4 138)(5 121)(6 140)(7 123)(8 142)(9 125)(10 144)(11 127)(12 110)(13 129)(14 112)(15 131)(16 114)(17 133)(18 116)(19 135)(20 118)(21 137)(22 120)(23 139)(24 122)(25 141)(26 124)(27 143)(28 126)(29 109)(30 128)(31 111)(32 130)(33 113)(34 132)(35 115)(36 134)(37 79)(38 98)(39 81)(40 100)(41 83)(42 102)(43 85)(44 104)(45 87)(46 106)(47 89)(48 108)(49 91)(50 74)(51 93)(52 76)(53 95)(54 78)(55 97)(56 80)(57 99)(58 82)(59 101)(60 84)(61 103)(62 86)(63 105)(64 88)(65 107)(66 90)(67 73)(68 92)(69 75)(70 94)(71 77)(72 96)
(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 18 19 36)(2 35 20 17)(3 16 21 34)(4 33 22 15)(5 14 23 32)(6 31 24 13)(7 12 25 30)(8 29 26 11)(9 10 27 28)(37 70 55 52)(38 51 56 69)(39 68 57 50)(40 49 58 67)(41 66 59 48)(42 47 60 65)(43 64 61 46)(44 45 62 63)(53 54 71 72)(73 100 91 82)(74 81 92 99)(75 98 93 80)(76 79 94 97)(77 96 95 78)(83 90 101 108)(84 107 102 89)(85 88 103 106)(86 105 104 87)(109 124 127 142)(110 141 128 123)(111 122 129 140)(112 139 130 121)(113 120 131 138)(114 137 132 119)(115 118 133 136)(116 135 134 117)(125 144 143 126)```

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 D9 D18 D18 D18 2- 1+4 Q8○D12 D4.10D18 kernel D4.10D18 C2×Dic18 D36⋊5C2 D4⋊2D9 Q8×D9 C9×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C9 C3 C1 # reps 1 3 3 6 2 1 1 3 3 1 3 9 9 3 1 2 6

Matrix representation of D4.10D18 in GL4(𝔽37) generated by

 0 0 1 0 0 0 0 1 36 0 0 0 0 36 0 0
,
 36 2 14 9 35 1 28 23 14 9 1 35 28 23 2 36
,
 19 2 4 16 35 21 21 20 4 16 18 35 21 20 2 16
,
 35 18 21 33 16 2 17 16 21 33 2 19 17 16 21 35
`G:=sub<GL(4,GF(37))| [0,0,36,0,0,0,0,36,1,0,0,0,0,1,0,0],[36,35,14,28,2,1,9,23,14,28,1,2,9,23,35,36],[19,35,4,21,2,21,16,20,4,21,18,2,16,20,35,16],[35,16,21,17,18,2,33,16,21,17,2,21,33,16,19,35] >;`

D4.10D18 in GAP, Magma, Sage, TeX

`D_4._{10}D_{18}`
`% in TeX`

`G:=Group("D4.10D18");`
`// GroupNames label`

`G:=SmallGroup(288,364);`
`// by ID`

`G=gap.SmallGroup(288,364);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,80,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=1,c^18=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^17>;`
`// generators/relations`

׿
×
𝔽