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## G = A4×C2×C20order 480 = 25·3·5

### Direct product of C2×C20 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C2×C20
 Chief series C1 — C22 — C23 — C22×C10 — C10×A4 — A4×C2×C10 — A4×C2×C20
 Lower central C22 — A4×C2×C20
 Upper central C1 — C2×C20

Generators and relations for A4×C2×C20
G = < a,b,c,d,e | a2=b20=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 316 in 132 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C12, A4, C2×C6, C15, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C12, C2×A4, C2×A4, C30, C23×C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C4×A4, C22×A4, C60, C5×A4, C2×C30, C22×C20, C22×C20, C23×C10, C2×C4×A4, C2×C60, C10×A4, C10×A4, C23×C20, A4×C20, A4×C2×C10, A4×C2×C20
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, C10, C12, A4, C2×C6, C15, C20, C2×C10, C2×C12, C2×A4, C30, C2×C20, C4×A4, C22×A4, C60, C5×A4, C2×C30, C2×C4×A4, C2×C60, C10×A4, A4×C20, A4×C2×C10, A4×C2×C20

Smallest permutation representation of A4×C2×C20
On 120 points
Generators in S120
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 113)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 101)(57 102)(58 103)(59 104)(60 105)
(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)
(21 99)(22 100)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 116)(42 117)(43 118)(44 119)(45 120)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 99)(22 100)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)
(1 53 100)(2 54 81)(3 55 82)(4 56 83)(5 57 84)(6 58 85)(7 59 86)(8 60 87)(9 41 88)(10 42 89)(11 43 90)(12 44 91)(13 45 92)(14 46 93)(15 47 94)(16 48 95)(17 49 96)(18 50 97)(19 51 98)(20 52 99)(21 80 107)(22 61 108)(23 62 109)(24 63 110)(25 64 111)(26 65 112)(27 66 113)(28 67 114)(29 68 115)(30 69 116)(31 70 117)(32 71 118)(33 72 119)(34 73 120)(35 74 101)(36 75 102)(37 76 103)(38 77 104)(39 78 105)(40 79 106)

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

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

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

160 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A ··· 6F 10A ··· 10L 10M ··· 10AB 12A ··· 12H 15A ··· 15H 20A ··· 20P 20Q ··· 20AF 30A ··· 30X 60A ··· 60AF order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 5 5 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 3 3 3 3 4 4 1 1 1 1 3 3 3 3 1 1 1 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C4 C5 C6 C6 C10 C10 C12 C15 C20 C30 C30 C60 A4 C2×A4 C2×A4 C4×A4 C5×A4 C10×A4 C10×A4 A4×C20 kernel A4×C2×C20 A4×C20 A4×C2×C10 C23×C20 C10×A4 C2×C4×A4 C22×C20 C23×C10 C4×A4 C22×A4 C22×C10 C23×C4 C2×A4 C22×C4 C24 C23 C2×C20 C20 C2×C10 C10 C2×C4 C4 C22 C2 # reps 1 2 1 2 4 4 4 2 8 4 8 8 16 16 8 32 1 2 1 4 4 8 4 16

Matrix representation of A4×C2×C20 in GL4(𝔽61) generated by

 60 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 53 0 0 0 0 53 0 0 0 0 53
,
 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 60 0 0 0 0 60 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,53,0,0,0,0,53,0,0,0,0,53],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,60,0,0,0,0,60,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

A4×C2×C20 in GAP, Magma, Sage, TeX

A_4\times C_2\times C_{20}
% in TeX

G:=Group("A4xC2xC20");
// GroupNames label

G:=SmallGroup(480,1126);
// by ID

G=gap.SmallGroup(480,1126);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,2,428,2539,4430]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^20=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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