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## G = D36.C4order 288 = 25·32

### The non-split extension by D36 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D36.C4
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — D36⋊5C2 — D36.C4
 Lower central C9 — C18 — D36.C4
 Upper central C1 — C4 — M4(2)

Generators and relations for D36.C4
G = < a,b,c | a36=b2=1, c4=a18, bab=a-1, cac-1=a19, cbc-1=a18b >

Subgroups: 332 in 93 conjugacy classes, 46 normal (28 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, M4(2), 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, C2×C3⋊C8, C3×M4(2), C4○D12, C9⋊C8, C72, Dic18, C4×D9, D36, C9⋊D4, C2×C36, D12.C4, C8×D9, C8⋊D9, C2×C9⋊C8, C9×M4(2), D365C2, D36.C4
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, D12.C4, C2×C4×D9, D36.C4

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 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 D12.C4 D36.C4 kernel D36.C4 C8×D9 C8⋊D9 C2×C9⋊C8 C9×M4(2) D36⋊5C2 Dic18 D36 C9⋊D4 C3×M4(2) C24 C2×C12 M4(2) 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 2 6

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

 70 31 0 0 42 28 0 0 0 0 0 51 0 0 10 0
,
 72 1 0 0 0 1 0 0 0 0 72 0 0 0 0 1
,
 27 0 0 0 0 27 0 0 0 0 0 1 0 0 27 0
`G:=sub<GL(4,GF(73))| [70,42,0,0,31,28,0,0,0,0,0,10,0,0,51,0],[72,0,0,0,1,1,0,0,0,0,72,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,0,27,0,0,1,0] >;`

D36.C4 in GAP, Magma, Sage, TeX

`D_{36}.C_4`
`% in TeX`

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

`G:=SmallGroup(288,117);`
`// by ID`

`G=gap.SmallGroup(288,117);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,80,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=b^2=1,c^4=a^18,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^18*b>;`
`// generators/relations`

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