Copied to
clipboard

G = A4⋊Dic10order 480 = 25·3·5

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

non-abelian, soluble, monomial

Aliases: A41Dic10, Dic5.3S4, (C5×A4)⋊Q8, A4⋊C4.D5, C51(A4⋊Q8), (C2×C10)⋊Dic6, C22⋊(C15⋊Q8), C10.9(C2×S4), C2.12(D5×S4), (C2×A4).1D10, C23.1(S3×D5), A4⋊Dic5.2C2, (A4×Dic5).1C2, (C22×C10).1D6, (C10×A4).1C22, (C22×Dic5).1S3, (C5×A4⋊C4).1C2, SmallGroup(480,975)

Series: Derived Chief Lower central Upper central

C1C22C10×A4 — A4⋊Dic10
C1C22C2×C10C5×A4C10×A4A4×Dic5 — A4⋊Dic10
C5×A4C10×A4 — A4⋊Dic10
C1C2

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 2A2B2C 3 4A4B4C4D4E4F5A5B 6 10A10B10C10D10E10F12A12B15A15B20A···20H30A30B
order122234444445561010101010101212151520···203030
size113381012123060602282266664040161612···121616

34 irreducible representations

dim111122222223344666
type+++++-+++--+++--+-
imageC1C2C2C2S3Q8D5D6D10Dic6Dic10S4C2×S4S3×D5C15⋊Q8A4⋊Q8D5×S4A4⋊Dic10
kernelA4⋊Dic10C5×A4⋊C4A4⋊Dic5A4×Dic5C22×Dic5C5×A4A4⋊C4C22×C10C2×A4C2×C10A4Dic5C10C23C22C5C2C1
# reps111111212242222144

Matrix representation of A4⋊Dic10 in GL5(𝔽61)

10000
01000
00001
00606060
00100
,
10000
01000
00606060
00001
00010
,
10000
01000
00100
00606060
00010
,
1234000
3358000
006000
000060
000600
,
5530000
136000
006000
000600
000060

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

׿
×
𝔽