Copied to
clipboard

## G = D4.D18order 288 = 25·32

### 3rd non-split extension by D4 of D18 acting via D18/C18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — D4.D18
 Chief series C1 — C3 — C9 — C18 — C36 — Dic18 — C2×Dic18 — D4.D18
 Lower central C9 — C18 — C36 — D4.D18
 Upper central C1 — C2 — C2×C4 — C4○D4

Generators and relations for D4.D18
G = < a,b,c,d | a4=b2=c18=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 316 in 90 conjugacy classes, 38 normal (30 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×3], C9, Dic3 [×2], C12 [×2], C12, C2×C6, C2×C6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C18, C18 [×2], C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, Dic9 [×2], C36 [×2], C36, C2×C18, C2×C18, C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C2×Dic6, C3×C4○D4, C9⋊C8 [×2], Dic18 [×2], Dic18, C2×Dic9, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, Q8.14D6, C4.Dic9, D4.D9 [×2], C9⋊Q16 [×2], C2×Dic18, C9×C4○D4, D4.D18
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C3⋊D4 [×2], C22×S3, C8.C22, D18 [×3], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, Q8.14D6, C2×C9⋊D4, D4.D18

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

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 9A 9B 9C 12A 12B 12C 12D 12E 18A 18B 18C 18D ··· 18L 36A ··· 36F 36G ··· 36O order 1 2 2 2 3 4 4 4 4 4 6 6 6 6 8 8 9 9 9 12 12 12 12 12 18 18 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 2 4 2 2 2 4 36 36 2 4 4 4 36 36 2 2 2 2 2 4 4 4 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D9 C3⋊D4 C3⋊D4 D18 D18 D18 C9⋊D4 C9⋊D4 C8.C22 Q8.14D6 D4.D18 kernel D4.D18 C4.Dic9 D4.D9 C9⋊Q16 C2×Dic18 C9×C4○D4 C3×C4○D4 C36 C2×C18 C2×C12 C3×D4 C3×Q8 C4○D4 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C9 C3 C1 # reps 1 1 2 2 1 1 1 1 1 1 1 1 3 2 2 3 3 3 6 6 1 2 6

Matrix representation of D4.D18 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 32 1 41 66 0 0 72 0 0 0 0 0 30 42 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 41 72 32 7 0 0 72 0 0 0 0 0 0 0 0 1
,
 31 28 0 0 0 0 45 3 0 0 0 0 0 0 68 66 71 49 0 0 63 59 17 25 0 0 7 7 0 0 0 0 49 5 35 19
,
 54 25 0 0 0 0 44 19 0 0 0 0 0 0 2 9 6 5 0 0 63 65 27 28 0 0 6 0 67 68 0 0 39 40 0 12

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,72,30,0,0,0,1,0,42,0,0,1,41,0,0,0,0,0,66,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,41,72,0,0,0,0,72,0,0,0,0,72,32,0,0,0,0,0,7,0,1],[31,45,0,0,0,0,28,3,0,0,0,0,0,0,68,63,7,49,0,0,66,59,7,5,0,0,71,17,0,35,0,0,49,25,0,19],[54,44,0,0,0,0,25,19,0,0,0,0,0,0,2,63,6,39,0,0,9,65,0,40,0,0,6,27,67,0,0,0,5,28,68,12] >;`

D4.D18 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(288,159);`
`// by ID`

`G=gap.SmallGroup(288,159);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,675,185,80,6725,292,9414]);`
`// Polycyclic`

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

׿
×
𝔽