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## G = Q8.C36order 288 = 25·32

### The non-split extension by Q8 of C36 acting via C36/C12=C3

Aliases: Q8.C36, C24.2A4, C8○D4⋊C9, Q8⋊C9.C4, C6.6(C4×A4), C8.(C3.A4), C3.(C8.A4), C12.15(C2×A4), C4○D4.2C18, (C3×Q8).3C12, Q8.C18.3C2, (C3×C8○D4).C3, C4.5(C2×C3.A4), C2.3(C4×C3.A4), (C3×C4○D4).6C6, SmallGroup(288,77)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8.C36
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×C4○D4 — Q8.C18 — Q8.C36
 Lower central Q8 — Q8.C36
 Upper central C1 — C24

Generators and relations for Q8.C36
G = < a,b,c | a4=1, b2=c36=a2, bab-1=a-1, cac-1=b, cbc-1=ab >

Smallest permutation representation of Q8.C36
On 144 points
Generators in S144
```(1 93 37 129)(2 56 38 20)(3 113 39 77)(4 96 40 132)(5 59 41 23)(6 116 42 80)(7 99 43 135)(8 62 44 26)(9 119 45 83)(10 102 46 138)(11 65 47 29)(12 122 48 86)(13 105 49 141)(14 68 50 32)(15 125 51 89)(16 108 52 144)(17 71 53 35)(18 128 54 92)(19 111 55 75)(21 131 57 95)(22 114 58 78)(24 134 60 98)(25 117 61 81)(27 137 63 101)(28 120 64 84)(30 140 66 104)(31 123 67 87)(33 143 69 107)(34 126 70 90)(36 74 72 110)(73 91 109 127)(76 94 112 130)(79 97 115 133)(82 100 118 136)(85 103 121 139)(88 106 124 142)
(1 55 37 19)(2 112 38 76)(3 95 39 131)(4 58 40 22)(5 115 41 79)(6 98 42 134)(7 61 43 25)(8 118 44 82)(9 101 45 137)(10 64 46 28)(11 121 47 85)(12 104 48 140)(13 67 49 31)(14 124 50 88)(15 107 51 143)(16 70 52 34)(17 127 53 91)(18 110 54 74)(20 130 56 94)(21 113 57 77)(23 133 59 97)(24 116 60 80)(26 136 62 100)(27 119 63 83)(29 139 65 103)(30 122 66 86)(32 142 68 106)(33 125 69 89)(35 73 71 109)(36 128 72 92)(75 93 111 129)(78 96 114 132)(81 99 117 135)(84 102 120 138)(87 105 123 141)(90 108 126 144)
(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)```

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 3 type + + + + image C1 C2 C3 C4 C6 C9 C12 C18 C36 C8.A4 Q8.C36 A4 C2×A4 C3.A4 C4×A4 C2×C3.A4 C4×C3.A4 kernel Q8.C36 Q8.C18 C3×C8○D4 Q8⋊C9 C3×C4○D4 C8○D4 C3×Q8 C4○D4 Q8 C3 C1 C24 C12 C8 C6 C4 C2 # reps 1 1 2 2 2 6 4 6 12 12 24 1 1 2 2 2 4

Matrix representation of Q8.C36 in GL2(𝔽73) generated by

 0 72 1 0
,
 46 0 0 27
,
 72 1 46 46
`G:=sub<GL(2,GF(73))| [0,1,72,0],[46,0,0,27],[72,46,1,46] >;`

Q8.C36 in GAP, Magma, Sage, TeX

`Q_8.C_{36}`
`% in TeX`

`G:=Group("Q8.C36");`
`// GroupNames label`

`G:=SmallGroup(288,77);`
`// by ID`

`G=gap.SmallGroup(288,77);`
`# by ID`

`G:=PCGroup([7,-2,-3,-2,-3,-2,2,-2,42,92,520,1271,172,2280,285,124]);`
`// Polycyclic`

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

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