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## G = C22.4D36order 288 = 25·32

### 3rd non-split extension by C22 of D36 acting via D36/D18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C22.4D36
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×C9⋊D4 — C22.4D36
 Lower central C9 — C2×C18 — C22.4D36
 Upper central C1 — C22 — C22⋊C4

Generators and relations for C22.4D36
G = < a,b,c,d | a2=b2=c36=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 524 in 117 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], S3, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C9, Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, D9, C18, C18 [×2], C18 [×2], C2×Dic3 [×5], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C22.D4, Dic9 [×3], C36 [×2], D18 [×3], C2×C18, C2×C18 [×2], C2×C18 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C2×Dic9, C2×Dic9 [×2], C2×Dic9 [×2], C9⋊D4 [×2], C2×C36 [×2], C22×D9, C22×C18, C23.21D6, C4⋊Dic9 [×2], D18⋊C4 [×2], C9×C22⋊C4, C22×Dic9, C2×C9⋊D4, C22.4D36
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, D12 [×2], C22×S3, C22.D4, D18 [×3], C2×D12, D42S3 [×2], D36 [×2], C22×D9, C23.21D6, C2×D36, D42D9 [×2], C22.4D36

Smallest permutation representation of C22.4D36
On 144 points
Generators in S144
```(1 126)(2 43)(3 128)(4 45)(5 130)(6 47)(7 132)(8 49)(9 134)(10 51)(11 136)(12 53)(13 138)(14 55)(15 140)(16 57)(17 142)(18 59)(19 144)(20 61)(21 110)(22 63)(23 112)(24 65)(25 114)(26 67)(27 116)(28 69)(29 118)(30 71)(31 120)(32 37)(33 122)(34 39)(35 124)(36 41)(38 90)(40 92)(42 94)(44 96)(46 98)(48 100)(50 102)(52 104)(54 106)(56 108)(58 74)(60 76)(62 78)(64 80)(66 82)(68 84)(70 86)(72 88)(73 141)(75 143)(77 109)(79 111)(81 113)(83 115)(85 117)(87 119)(89 121)(91 123)(93 125)(95 127)(97 129)(99 131)(101 133)(103 135)(105 137)(107 139)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 141)(58 142)(59 143)(60 144)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)
(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 93 94 36)(2 35 95 92)(3 91 96 34)(4 33 97 90)(5 89 98 32)(6 31 99 88)(7 87 100 30)(8 29 101 86)(9 85 102 28)(10 27 103 84)(11 83 104 26)(12 25 105 82)(13 81 106 24)(14 23 107 80)(15 79 108 22)(16 21 73 78)(17 77 74 20)(18 19 75 76)(37 130 121 46)(38 45 122 129)(39 128 123 44)(40 43 124 127)(41 126 125 42)(47 120 131 72)(48 71 132 119)(49 118 133 70)(50 69 134 117)(51 116 135 68)(52 67 136 115)(53 114 137 66)(54 65 138 113)(55 112 139 64)(56 63 140 111)(57 110 141 62)(58 61 142 109)(59 144 143 60)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 D9 D12 D18 D18 D36 D4⋊2S3 D4⋊2D9 kernel C22.4D36 C4⋊Dic9 D18⋊C4 C9×C22⋊C4 C22×Dic9 C2×C9⋊D4 C3×C22⋊C4 C2×C18 C2×C12 C22×C6 C18 C22⋊C4 C2×C6 C2×C4 C23 C22 C6 C2 # reps 1 2 2 1 1 1 1 2 2 1 4 3 4 6 3 12 2 6

Matrix representation of C22.4D36 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 31 25 0 0 6 6
,
 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 25 29 0 0 8 33 0 0 0 0 36 35 0 0 0 1
,
 27 32 0 0 5 10 0 0 0 0 36 35 0 0 1 1
`G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,31,6,0,0,25,6],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[25,8,0,0,29,33,0,0,0,0,36,0,0,0,35,1],[27,5,0,0,32,10,0,0,0,0,36,1,0,0,35,1] >;`

C22.4D36 in GAP, Magma, Sage, TeX

`C_2^2._4D_{36}`
`% in TeX`

`G:=Group("C2^2.4D36");`
`// GroupNames label`

`G:=SmallGroup(288,96);`
`// by ID`

`G=gap.SmallGroup(288,96);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,142,6725,292,9414]);`
`// Polycyclic`

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

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