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## G = 2- 1+4⋊C9order 288 = 25·32

### The semidirect product of 2- 1+4 and C9 acting via C9/C3=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — 2- 1+4⋊C9
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — C2×Q8⋊C9 — 2- 1+4⋊C9
 Lower central Q8 — 2- 1+4⋊C9
 Upper central C1 — C6 — C3×D4

Generators and relations for 2- 1+4⋊C9
G = < a,b,c,d,e | a4=b2=e9=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece-1=d, ede-1=cd >

Subgroups: 201 in 79 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, D4, D4, Q8, Q8, C9, C12, C12, C2×C6, C2×C6, C2×Q8, C2×Q8, C4○D4, C4○D4, C18, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, 2- 1+4, C36, C2×C18, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, Q8⋊C9, D4×C9, C3×2- 1+4, C2×Q8⋊C9, Q8.C18, 2- 1+4⋊C9
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C3.A4, C2×C18, C22×A4, C2×C3.A4, D4.A4, C22×C3.A4, 2- 1+4⋊C9

Smallest permutation representation of 2- 1+4⋊C9
On 144 points
Generators in S144
```(1 98 85 110)(2 99 86 111)(3 91 87 112)(4 92 88 113)(5 93 89 114)(6 94 90 115)(7 95 82 116)(8 96 83 117)(9 97 84 109)(10 23 70 58)(11 24 71 59)(12 25 72 60)(13 26 64 61)(14 27 65 62)(15 19 66 63)(16 20 67 55)(17 21 68 56)(18 22 69 57)(28 49 136 37)(29 50 137 38)(30 51 138 39)(31 52 139 40)(32 53 140 41)(33 54 141 42)(34 46 142 43)(35 47 143 44)(36 48 144 45)(73 126 127 107)(74 118 128 108)(75 119 129 100)(76 120 130 101)(77 121 131 102)(78 122 132 103)(79 123 133 104)(80 124 134 105)(81 125 135 106)
(1 51)(2 52)(3 53)(4 54)(5 46)(6 47)(7 48)(8 49)(9 50)(10 135)(11 127)(12 128)(13 129)(14 130)(15 131)(16 132)(17 133)(18 134)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 118)(26 119)(27 120)(28 96)(29 97)(30 98)(31 99)(32 91)(33 92)(34 93)(35 94)(36 95)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 82)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 100)(62 101)(63 102)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 73)(72 74)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)(115 143)(116 144)(117 136)
(1 129 85 75)(2 111 86 99)(3 102 87 121)(4 132 88 78)(5 114 89 93)(6 105 90 124)(7 135 82 81)(8 117 83 96)(9 108 84 118)(10 45 70 48)(11 59 71 24)(12 29 72 137)(13 39 64 51)(14 62 65 27)(15 32 66 140)(16 42 67 54)(17 56 68 21)(18 35 69 143)(19 53 63 41)(20 33 55 141)(22 47 57 44)(23 36 58 144)(25 50 60 38)(26 30 61 138)(28 49 136 37)(31 52 139 40)(34 46 142 43)(73 126 127 107)(74 109 128 97)(76 120 130 101)(77 112 131 91)(79 123 133 104)(80 115 134 94)(92 103 113 122)(95 106 116 125)(98 100 110 119)
(1 110 85 98)(2 101 86 120)(3 131 87 77)(4 113 88 92)(5 104 89 123)(6 134 90 80)(7 116 82 95)(8 107 83 126)(9 128 84 74)(10 58 70 23)(11 28 71 136)(12 38 72 50)(13 61 64 26)(14 31 65 139)(15 41 66 53)(16 55 67 20)(17 34 68 142)(18 44 69 47)(19 32 63 140)(21 46 56 43)(22 35 57 143)(24 49 59 37)(25 29 60 137)(27 52 62 40)(30 51 138 39)(33 54 141 42)(36 48 144 45)(73 117 127 96)(75 119 129 100)(76 111 130 99)(78 122 132 103)(79 114 133 93)(81 125 135 106)(91 102 112 121)(94 105 115 124)(97 108 109 118)
(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)```

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 4 4 4 type + + + + + + - image C1 C2 C2 C3 C6 C6 C9 C18 C18 A4 C2×A4 C2×A4 C3.A4 C2×C3.A4 C2×C3.A4 D4.A4 D4.A4 2- 1+4⋊C9 kernel 2- 1+4⋊C9 C2×Q8⋊C9 Q8.C18 C3×2- 1+4 C6×Q8 C3×C4○D4 2- 1+4 C2×Q8 C4○D4 C3×D4 C12 C2×C6 D4 C4 C22 C3 C3 C1 # reps 1 2 1 2 4 2 6 12 6 1 1 2 2 2 4 1 2 6

Matrix representation of 2- 1+4⋊C9 in GL7(𝔽37)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 36 0 0 0 0 0 0 36 0 0 0 36 1 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 23 14 32 0 0 0 0 9 14 27 5 0 0 0 5 0 23 14 0 0 0 10 32 9 14
,
 36 0 0 0 0 0 0 0 36 0 0 0 0 0 9 7 1 0 0 0 0 0 0 0 36 1 0 0 0 0 0 35 1 0 0 0 0 0 0 0 1 36 0 0 0 0 0 2 36
,
 1 0 0 0 0 0 0 24 36 0 0 0 0 0 18 0 36 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 1 0 0 0 0 0 0 0 1 0 0
,
 26 4 0 0 0 0 0 7 4 35 0 0 0 0 33 5 7 0 0 0 0 0 0 0 0 32 10 32 0 0 0 10 27 10 0 0 0 0 0 32 27 5 0 0 0 10 27 27 0

`G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,36,36,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,23,9,5,10,0,0,0,14,14,0,32,0,0,0,32,27,23,9,0,0,0,0,5,14,14],[36,0,9,0,0,0,0,0,36,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,35,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,2,0,0,0,0,0,36,36],[1,24,18,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,36,0,0,0,0,0,0,0,36,0,0],[26,7,33,0,0,0,0,4,4,5,0,0,0,0,0,35,7,0,0,0,0,0,0,0,0,10,0,10,0,0,0,32,27,32,27,0,0,0,10,10,27,27,0,0,0,32,0,5,0] >;`

2- 1+4⋊C9 in GAP, Magma, Sage, TeX

`2_-^{1+4}\rtimes C_9`
`% in TeX`

`G:=Group("ES-(2,2):C9");`
`// GroupNames label`

`G:=SmallGroup(288,349);`
`// by ID`

`G=gap.SmallGroup(288,349);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,79,648,172,1153,285,124]);`
`// Polycyclic`

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

׿
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