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## G = A4×C36order 432 = 24·33

### Direct product of C36 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C36
 Chief series C1 — C22 — C2×C6 — C22×C6 — C22×C18 — A4×C18 — A4×C36
 Lower central C22 — A4×C36
 Upper central C1 — C36

Generators and relations for A4×C36
G = < a,b,c,d | a36=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Subgroups: 186 in 81 conjugacy classes, 39 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, C23, C9, C9, C32, C12, C12, A4, C2×C6, C2×C6, C22×C4, C18, C18, C3×C6, C2×C12, C2×A4, C22×C6, C3×C9, C36, C36, C3.A4, C2×C18, C2×C18, C3×C12, C3×A4, C4×A4, C22×C12, C3×C18, C2×C36, C2×C3.A4, C22×C18, C6×A4, C3×C36, C9×A4, C4×C3.A4, C22×C36, C12×A4, A4×C18, A4×C36
Quotients: C1, C2, C3, C4, C6, C9, C32, C12, A4, C18, C3×C6, C2×A4, C3×C9, C36, C3×C12, C3×A4, C4×A4, C3×C18, C6×A4, C3×C36, C9×A4, C12×A4, A4×C18, A4×C36

Smallest permutation representation of A4×C36
On 108 points
Generators in S108
(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)
(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)
(1 59 77)(2 60 78)(3 61 79)(4 62 80)(5 63 81)(6 64 82)(7 65 83)(8 66 84)(9 67 85)(10 68 86)(11 69 87)(12 70 88)(13 71 89)(14 72 90)(15 37 91)(16 38 92)(17 39 93)(18 40 94)(19 41 95)(20 42 96)(21 43 97)(22 44 98)(23 45 99)(24 46 100)(25 47 101)(26 48 102)(27 49 103)(28 50 104)(29 51 105)(30 52 106)(31 53 107)(32 54 108)(33 55 73)(34 56 74)(35 57 75)(36 58 76)

G:=sub<Sym(108)| (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), (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,59,77)(2,60,78)(3,61,79)(4,62,80)(5,63,81)(6,64,82)(7,65,83)(8,66,84)(9,67,85)(10,68,86)(11,69,87)(12,70,88)(13,71,89)(14,72,90)(15,37,91)(16,38,92)(17,39,93)(18,40,94)(19,41,95)(20,42,96)(21,43,97)(22,44,98)(23,45,99)(24,46,100)(25,47,101)(26,48,102)(27,49,103)(28,50,104)(29,51,105)(30,52,106)(31,53,107)(32,54,108)(33,55,73)(34,56,74)(35,57,75)(36,58,76)>;

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), (37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108), (1,59,77)(2,60,78)(3,61,79)(4,62,80)(5,63,81)(6,64,82)(7,65,83)(8,66,84)(9,67,85)(10,68,86)(11,69,87)(12,70,88)(13,71,89)(14,72,90)(15,37,91)(16,38,92)(17,39,93)(18,40,94)(19,41,95)(20,42,96)(21,43,97)(22,44,98)(23,45,99)(24,46,100)(25,47,101)(26,48,102)(27,49,103)(28,50,104)(29,51,105)(30,52,106)(31,53,107)(32,54,108)(33,55,73)(34,56,74)(35,57,75)(36,58,76) );

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)], [(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108)], [(1,59,77),(2,60,78),(3,61,79),(4,62,80),(5,63,81),(6,64,82),(7,65,83),(8,66,84),(9,67,85),(10,68,86),(11,69,87),(12,70,88),(13,71,89),(14,72,90),(15,37,91),(16,38,92),(17,39,93),(18,40,94),(19,41,95),(20,42,96),(21,43,97),(22,44,98),(23,45,99),(24,46,100),(25,47,101),(26,48,102),(27,49,103),(28,50,104),(29,51,105),(30,52,106),(31,53,107),(32,54,108),(33,55,73),(34,56,74),(35,57,75),(36,58,76)]])

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G ··· 6L 9A ··· 9F 9G ··· 9R 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12T 18A ··· 18F 18G ··· 18R 18S ··· 18AD 36A ··· 36L 36M ··· 36X 36Y ··· 36AV order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 6 6 6 6 ··· 6 9 ··· 9 9 ··· 9 12 12 12 12 12 12 12 12 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 3 3 1 1 4 ··· 4 1 1 3 3 1 1 3 3 3 3 4 ··· 4 1 ··· 1 4 ··· 4 1 1 1 1 3 3 3 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C3 C4 C6 C6 C6 C9 C12 C12 C12 C18 C36 A4 C2×A4 C3×A4 C4×A4 C6×A4 C9×A4 C12×A4 A4×C18 A4×C36 kernel A4×C36 A4×C18 C4×C3.A4 C22×C36 C12×A4 C9×A4 C2×C3.A4 C22×C18 C6×A4 C4×A4 C3.A4 C2×C18 C3×A4 C2×A4 A4 C36 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 4 2 2 2 4 2 2 18 8 4 4 18 36 1 1 2 2 2 6 4 6 12

Matrix representation of A4×C36 in GL3(𝔽37) generated by

 24 0 0 0 24 0 0 0 24
,
 1 0 0 26 36 0 10 0 36
,
 36 0 0 0 36 0 27 0 1
,
 26 35 0 0 11 1 0 27 0
G:=sub<GL(3,GF(37))| [24,0,0,0,24,0,0,0,24],[1,26,10,0,36,0,0,0,36],[36,0,27,0,36,0,0,0,1],[26,0,0,35,11,27,0,1,0] >;

A4×C36 in GAP, Magma, Sage, TeX

A_4\times C_{36}
% in TeX

G:=Group("A4xC36");
// GroupNames label

G:=SmallGroup(432,325);
// by ID

G=gap.SmallGroup(432,325);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-3,-2,2,126,142,4548,7951]);
// Polycyclic

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

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