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## G = C36.A4order 432 = 24·33

### The non-split extension by C36 of A4 acting via A4/C22=C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — C36.A4
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C9 — C18.A4 — C36.A4
 Lower central Q8 — C3×Q8 — C36.A4
 Upper central C1 — C12 — C36

Generators and relations for C36.A4
G = < a,b,c,d | a36=d3=1, b2=c2=a18, ab=ba, ac=ca, dad-1=a25, cbc-1=a18b, dbd-1=a18bc, dcd-1=b >

Smallest permutation representation of C36.A4
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 67 19 49)(2 68 20 50)(3 69 21 51)(4 70 22 52)(5 71 23 53)(6 72 24 54)(7 37 25 55)(8 38 26 56)(9 39 27 57)(10 40 28 58)(11 41 29 59)(12 42 30 60)(13 43 31 61)(14 44 32 62)(15 45 33 63)(16 46 34 64)(17 47 35 65)(18 48 36 66)(73 142 91 124)(74 143 92 125)(75 144 93 126)(76 109 94 127)(77 110 95 128)(78 111 96 129)(79 112 97 130)(80 113 98 131)(81 114 99 132)(82 115 100 133)(83 116 101 134)(84 117 102 135)(85 118 103 136)(86 119 104 137)(87 120 105 138)(88 121 106 139)(89 122 107 140)(90 123 108 141)
(1 138 19 120)(2 139 20 121)(3 140 21 122)(4 141 22 123)(5 142 23 124)(6 143 24 125)(7 144 25 126)(8 109 26 127)(9 110 27 128)(10 111 28 129)(11 112 29 130)(12 113 30 131)(13 114 31 132)(14 115 32 133)(15 116 33 134)(16 117 34 135)(17 118 35 136)(18 119 36 137)(37 75 55 93)(38 76 56 94)(39 77 57 95)(40 78 58 96)(41 79 59 97)(42 80 60 98)(43 81 61 99)(44 82 62 100)(45 83 63 101)(46 84 64 102)(47 85 65 103)(48 86 66 104)(49 87 67 105)(50 88 68 106)(51 89 69 107)(52 90 70 108)(53 91 71 73)(54 92 72 74)
(2 14 26)(3 27 15)(5 17 29)(6 30 18)(8 20 32)(9 33 21)(11 23 35)(12 36 24)(37 144 93)(38 121 82)(39 134 107)(40 111 96)(41 124 85)(42 137 74)(43 114 99)(44 127 88)(45 140 77)(46 117 102)(47 130 91)(48 143 80)(49 120 105)(50 133 94)(51 110 83)(52 123 108)(53 136 97)(54 113 86)(55 126 75)(56 139 100)(57 116 89)(58 129 78)(59 142 103)(60 119 92)(61 132 81)(62 109 106)(63 122 95)(64 135 84)(65 112 73)(66 125 98)(67 138 87)(68 115 76)(69 128 101)(70 141 90)(71 118 79)(72 131 104)```

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 3 3 3 6 type + + + + image C1 C2 C3 C3 C3 C6 C6 C6 C4.A4 C3×C4.A4 A4 C2×A4 3- 1+2 C3×A4 C2×3- 1+2 C6×A4 C9⋊A4 C2×C9⋊A4 C36.A4 kernel C36.A4 C18.A4 Q8.C18 C9×C4○D4 C3×C4.A4 Q8⋊C9 Q8×C9 C3×SL2(𝔽3) C9 C3 C36 C18 C4○D4 C12 Q8 C6 C4 C2 C1 # reps 1 1 4 2 2 4 2 2 6 12 1 1 2 2 2 2 6 6 4

Matrix representation of C36.A4 in GL5(𝔽37)

 6 0 0 0 0 0 6 0 0 0 0 0 20 24 13 0 0 5 2 32 0 0 18 19 15
,
 10 11 0 0 0 11 27 0 0 0 0 0 0 36 1 0 0 0 36 0 0 0 1 36 0
,
 0 36 0 0 0 1 0 0 0 0 0 0 0 1 36 0 0 1 0 36 0 0 0 0 36
,
 1 0 0 0 0 27 26 0 0 0 0 0 26 11 0 0 0 0 11 26 0 0 0 11 0

`G:=sub<GL(5,GF(37))| [6,0,0,0,0,0,6,0,0,0,0,0,20,5,18,0,0,24,2,19,0,0,13,32,15],[10,11,0,0,0,11,27,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[0,1,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,36,36,36],[1,27,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,11,11,11,0,0,0,26,0] >;`

C36.A4 in GAP, Magma, Sage, TeX

`C_{36}.A_4`
`% in TeX`

`G:=Group("C36.A4");`
`// GroupNames label`

`G:=SmallGroup(432,330);`
`// by ID`

`G=gap.SmallGroup(432,330);`
`# by ID`

`G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,261,79,1901,172,3414,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^36=d^3=1,b^2=c^2=a^18,a*b=b*a,a*c=c*a,d*a*d^-1=a^25,c*b*c^-1=a^18*b,d*b*d^-1=a^18*b*c,d*c*d^-1=b>;`
`// generators/relations`

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