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G = A4×C22×C10  order 480 = 25·3·5

Direct product of C22×C10 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: A4×C22×C10, C25⋊3C15, C24⋊5C30, (C24×C10)⋊1C3, C23⋊3(C2×C30), (C23×C10)⋊7C6, C22⋊(C22×C30), (C2×C10)⋊3(C22×C6), (C22×C10)⋊7(C2×C6), SmallGroup(480,1208)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — A4×C22×C10
Chief series
C1 — C22 — C2×C10 — C5×A4 — C10×A4 — A4×C2×C10 — A4×C22×C10
Lower central
C22 — A4×C22×C10

Subgroups: 908 in 356 conjugacy classes, 96 normal (12 characteristic)
C1, C2 [×7], C2 [×8], C3, C22, C22 [×7], C22 [×49], C5, C6 [×7], C23, C23 [×7], C23 [×49], C10 [×7], C10 [×8], A4, C2×C6 [×7], C15, C24 [×7], C24 [×8], C2×C10, C2×C10 [×7], C2×C10 [×49], C2×A4 [×7], C22×C6, C30 [×7], C25, C22×C10, C22×C10 [×7], C22×C10 [×49], C22×A4 [×7], C5×A4, C2×C30 [×7], C23×C10 [×7], C23×C10 [×8], C23×A4, C10×A4 [×7], C22×C30, C24×C10, A4×C2×C10 [×7], A4×C22×C10

Quotients:
C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], C23, C10 [×7], A4, C2×C6 [×7], C15, C2×C10 [×7], C2×A4 [×7], C22×C6, C30 [×7], C22×C10, C22×A4 [×7], C5×A4, C2×C30 [×7], C23×A4, C10×A4 [×7], C22×C30, A4×C2×C10 [×7], A4×C22×C10

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c10=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

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

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

(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)

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

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

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

Matrix representation ►G ⊆ GL5(𝔽31)

300000
01000
003000
000300
000030
,
10000
01000
003000
000300
000030
,
10000
030000
002700
000270
000027
,
10000
01000
003000
000300
002801
,
10000
01000
003000
00610
000030
,
250000
01000
0025290
00161
006280

G:=sub<GL(5,GF(31))| [30,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,30,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,28,0,0,0,30,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,30,6,0,0,0,0,1,0,0,0,0,0,30],[25,0,0,0,0,0,1,0,0,0,0,0,25,1,6,0,0,29,6,28,0,0,0,1,0] >;
 

160 conjugacy classes

class 1 2A···2G2H···2O3A3B5A5B5C5D6A···6N10A···10AB10AC···10BH15A···15H30A···30BD
order12···22···23355556···610···1010···1015···1530···30
size11···13···34411114···41···13···34···44···4

160 irreducible representations

dim111111113333
type++++
imageC1C2C3C5C6C10C15C30A4C2×A4C5×A4C10×A4
kernelA4×C22×C10A4×C2×C10C24×C10C23×A4C23×C10C22×A4C25C24C22×C10C2×C10C23C22
# reps1724142885617428

In GAP, Magma, Sage, TeX 

A_4\times C_2^2\times C_{10}
 
% in TeX
 
G:=Group("A4xC2^2xC10");
 
// GroupNames label
 
G:=SmallGroup(480,1208);
 
// by ID
 
G=gap.SmallGroup(480,1208);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,1286,2232]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^10=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
 
// generators/relations
 

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