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## G = A4⋊Dic10order 480 = 25·3·5

### The semidirect product of A4 and Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 548 in 84 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C22, C22 [×2], C5, C6, C2×C4 [×6], Q8 [×2], C23, C10, C10 [×2], Dic3 [×2], C12, A4, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5, Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×2], Dic6, C2×A4, C30, C22⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×C20 [×2], C22×C10, A4⋊C4, A4⋊C4, C4×A4, C5×Dic3, C3×Dic5, Dic15, C5×A4, C10.D4 [×2], C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, A4⋊Q8, C15⋊Q8, C10×A4, Dic5.14D4, C5×A4⋊C4, A4⋊Dic5, A4×Dic5, A4⋊Dic10
Quotients: C1, C2 [×3], C22, S3, Q8, D5, D6, D10, Dic6, S4, Dic10, C2×S4, S3×D5, A4⋊Q8, C15⋊Q8, D5×S4, A4⋊Dic10

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

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6 10A 10B 10C 10D 10E 10F 12A 12B 15A 15B 20A ··· 20H 30A 30B order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 10 10 10 10 10 10 12 12 15 15 20 ··· 20 30 30 size 1 1 3 3 8 10 12 12 30 60 60 2 2 8 2 2 6 6 6 6 40 40 16 16 12 ··· 12 16 16

34 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 3 3 4 4 6 6 6 type + + + + + - + + + - - + + + - - + - image C1 C2 C2 C2 S3 Q8 D5 D6 D10 Dic6 Dic10 S4 C2×S4 S3×D5 C15⋊Q8 A4⋊Q8 D5×S4 A4⋊Dic10 kernel A4⋊Dic10 C5×A4⋊C4 A4⋊Dic5 A4×Dic5 C22×Dic5 C5×A4 A4⋊C4 C22×C10 C2×A4 C2×C10 A4 Dic5 C10 C23 C22 C5 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 4 2 2 2 2 1 4 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 60 60 60 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 60 60 60 0 0 0 1 0
,
 12 34 0 0 0 33 58 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 60 0
,
 55 30 0 0 0 13 6 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60

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],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,60,0,1,0,0,60,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,60,0,0,0,0,60,1,0,0,0,60,0],[12,33,0,0,0,34,58,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,60,0],[55,13,0,0,0,30,6,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60] >;

A4⋊Dic10 in GAP, Magma, Sage, TeX

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

G:=Group("A4:Dic10");
// GroupNames label

G:=SmallGroup(480,975);
// by ID

G=gap.SmallGroup(480,975);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,28,85,36,234,3364,5052,1286,2953,2232]);
// 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=d*a*d^-1=a*b=b*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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