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## G = D36.2C4order 288 = 25·32

### The non-split extension by D36 of C4 acting through Inn(D36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D36.2C4
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — D36⋊5C2 — D36.2C4
 Lower central C9 — C18 — D36.2C4
 Upper central C1 — C8 — C2×C8

Generators and relations for D36.2C4
G = < a,b,c | a36=b2=1, c4=a18, bab=a-1, ac=ca, bc=cb >

Subgroups: 332 in 93 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), C4○D4, D9, C18, C18, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, Dic9, C36, D18, C2×C18, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C9⋊C8, C72, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C8○D12, C8×D9, C8⋊D9, C4.Dic9, C2×C72, D365C2, D36.2C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, D9, C4×S3, C22×S3, C8○D4, D18, S3×C2×C4, C4×D9, C22×D9, C8○D12, C2×C4×D9, D36.2C4

Smallest permutation representation of D36.2C4
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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(28 36)(29 35)(30 34)(31 33)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(70 72)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(86 108)(87 107)(88 106)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(109 133)(110 132)(111 131)(112 130)(113 129)(114 128)(115 127)(116 126)(117 125)(118 124)(119 123)(120 122)(134 144)(135 143)(136 142)(137 141)(138 140)
(1 40 102 126 19 58 84 144)(2 41 103 127 20 59 85 109)(3 42 104 128 21 60 86 110)(4 43 105 129 22 61 87 111)(5 44 106 130 23 62 88 112)(6 45 107 131 24 63 89 113)(7 46 108 132 25 64 90 114)(8 47 73 133 26 65 91 115)(9 48 74 134 27 66 92 116)(10 49 75 135 28 67 93 117)(11 50 76 136 29 68 94 118)(12 51 77 137 30 69 95 119)(13 52 78 138 31 70 96 120)(14 53 79 139 32 71 97 121)(15 54 80 140 33 72 98 122)(16 55 81 141 34 37 99 123)(17 56 82 142 35 38 100 124)(18 57 83 143 36 39 101 125)```

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 24A ··· 24H 36A ··· 36L 72A ··· 72X order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 8 8 9 9 9 12 12 12 12 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 2 18 18 2 1 1 2 18 18 2 2 2 1 1 1 1 2 2 18 18 18 18 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 D9 C4×S3 C4×S3 C8○D4 D18 D18 C4×D9 C4×D9 C8○D12 D36.2C4 kernel D36.2C4 C8×D9 C8⋊D9 C4.Dic9 C2×C72 D36⋊5C2 Dic18 D36 C9⋊D4 C2×C24 C24 C2×C12 C2×C8 C12 C2×C6 C9 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 2 2 4 1 2 1 3 2 2 4 6 3 6 6 8 24

Matrix representation of D36.2C4 in GL4(𝔽73) generated by

 66 7 0 0 66 59 0 0 0 0 31 45 0 0 28 3
,
 1 1 0 0 0 72 0 0 0 0 72 1 0 0 0 1
,
 51 0 0 0 0 51 0 0 0 0 72 0 0 0 0 72
`G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,31,28,0,0,45,3],[1,0,0,0,1,72,0,0,0,0,72,0,0,0,1,1],[51,0,0,0,0,51,0,0,0,0,72,0,0,0,0,72] >;`

D36.2C4 in GAP, Magma, Sage, TeX

`D_{36}._2C_4`
`% in TeX`

`G:=Group("D36.2C4");`
`// GroupNames label`

`G:=SmallGroup(288,112);`
`// by ID`

`G=gap.SmallGroup(288,112);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,80,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^2=1,c^4=a^18,b*a*b=a^-1,a*c=c*a,b*c=c*b>;`
`// generators/relations`

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