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## G = A4×Dic10order 480 = 25·3·5

### Direct product of A4 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — A4×Dic10
 Chief series C1 — C5 — C2×C10 — C22×C10 — C10×A4 — A4×Dic5 — A4×Dic10
 Lower central C2×C10 — C22×C10 — A4×Dic10
 Upper central C1 — C2 — C4

Generators and relations for A4×Dic10
G = < a,b,c,d,e | a2=b2=c3=d20=1, e2=d10, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 480 in 92 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, Q8, C23, C10, C10, C12, A4, C15, C22×C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C3×Q8, C2×A4, C30, C22×Q8, Dic10, Dic10, C2×Dic5, C2×C20, C22×C10, C4×A4, C4×A4, C3×Dic5, C60, C5×A4, C2×Dic10, C22×Dic5, C22×C20, Q8×A4, C3×Dic10, C10×A4, C22×Dic10, A4×Dic5, A4×C20, A4×Dic10
Quotients: C1, C2, C3, C22, C6, Q8, D5, A4, C2×C6, D10, C3×Q8, C2×A4, C3×D5, Dic10, C22×A4, C6×D5, Q8×A4, C3×Dic10, D5×A4, C2×D5×A4, A4×Dic10

Smallest permutation representation of A4×Dic10
On 120 points
Generators in S120
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(1 34 64)(2 35 65)(3 36 66)(4 37 67)(5 38 68)(6 39 69)(7 40 70)(8 21 71)(9 22 72)(10 23 73)(11 24 74)(12 25 75)(13 26 76)(14 27 77)(15 28 78)(16 29 79)(17 30 80)(18 31 61)(19 32 62)(20 33 63)(41 115 97)(42 116 98)(43 117 99)(44 118 100)(45 119 81)(46 120 82)(47 101 83)(48 102 84)(49 103 85)(50 104 86)(51 105 87)(52 106 88)(53 107 89)(54 108 90)(55 109 91)(56 110 92)(57 111 93)(58 112 94)(59 113 95)(60 114 96)
(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 43 11 53)(2 42 12 52)(3 41 13 51)(4 60 14 50)(5 59 15 49)(6 58 16 48)(7 57 17 47)(8 56 18 46)(9 55 19 45)(10 54 20 44)(21 110 31 120)(22 109 32 119)(23 108 33 118)(24 107 34 117)(25 106 35 116)(26 105 36 115)(27 104 37 114)(28 103 38 113)(29 102 39 112)(30 101 40 111)(61 82 71 92)(62 81 72 91)(63 100 73 90)(64 99 74 89)(65 98 75 88)(66 97 76 87)(67 96 77 86)(68 95 78 85)(69 94 79 84)(70 93 80 83)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 3 3 4 4 4 4 4 4 5 5 6 6 10 10 10 10 10 10 12 12 12 12 12 12 15 15 15 15 20 20 20 20 20 20 20 20 30 30 30 30 60 ··· 60 size 1 1 3 3 4 4 2 6 10 10 30 30 2 2 4 4 2 2 6 6 6 6 8 8 40 40 40 40 8 8 8 8 2 2 2 2 6 6 6 6 8 8 8 8 8 ··· 8

52 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 6 6 6 6 type + + + - + + - + + + - + + - image C1 C2 C2 C3 C6 C6 Q8 D5 D10 C3×Q8 C3×D5 Dic10 C6×D5 C3×Dic10 A4 C2×A4 C2×A4 Q8×A4 D5×A4 C2×D5×A4 A4×Dic10 kernel A4×Dic10 A4×Dic5 A4×C20 C22×Dic10 C22×Dic5 C22×C20 C5×A4 C4×A4 C2×A4 C2×C10 C22×C4 A4 C23 C22 Dic10 Dic5 C20 C5 C4 C2 C1 # reps 1 2 1 2 4 2 1 2 2 2 4 4 4 8 1 2 1 1 2 2 4

Matrix representation of A4×Dic10 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 60 60 60 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 60 60 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 47 0 0 0 0 0 0 47 0 0 14 14 14
,
 27 31 0 0 0 31 4 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 56 14 0 0 0 33 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,60,0,0,0,1,60,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,1,0,60,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,47,0,14,0,0,0,0,14,0,0,0,47,14],[27,31,0,0,0,31,4,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[56,33,0,0,0,14,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×Dic10 in GAP, Magma, Sage, TeX

A_4\times {\rm Dic}_{10}
% in TeX

G:=Group("A4xDic10");
// GroupNames label

G:=SmallGroup(480,1035);
// by ID

G=gap.SmallGroup(480,1035);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,84,197,92,648,271,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^20=1,e^2=d^10,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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