Copied to
clipboard

?

G = D20⋊25D6  order 480 = 25·3·5

8th semidirect product of D20 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20⋊25D6, D12⋊25D10, Dic6⋊23D10, D60⋊34C22, C30.21C24, D30.7C23, C15⋊12+ (1+4), C60.164C23, Dic30⋊31C22, Dic15.10C23, (C2×C20)⋊8D6, C4○D12⋊8D5, (C6×D20)⋊3C2, (C4×S3)⋊1D10, (C2×C12)⋊8D10, (S3×D20)⋊11C2, (C2×D20)⋊14S3, C3⋊D4⋊13D10, C5⋊1(D4⋊6D6), (C2×C60)⋊4C22, (C22×D5)⋊5D6, C20⋊D6⋊11C2, C3⋊1(D4⋊8D10), C3⋊D20⋊1C22, C15⋊D4⋊1C22, D20⋊S3⋊12C2, D20⋊5S3⋊12C2, D60⋊11C2⋊7C2, (C6×D5).7C23, D6.7(C22×D5), C6.21(C23×D5), (S3×C20)⋊10C22, (C4×D15)⋊12C22, (C5×D12)⋊32C22, (C3×D20)⋊25C22, C15⋊7D4⋊15C22, (S3×C10).7C23, C10.21(S3×C23), (D5×Dic3)⋊1C22, D10.7(C22×S3), (C2×C30).240C23, C20.130(C22×S3), (C5×Dic6)⋊29C22, C12.129(C22×D5), (C5×Dic3).10C23, Dic3.10(C22×D5), (C2×C4)⋊4(S3×D5), (D5×C3⋊D4)⋊1C2, C4.135(C2×S3×D5), (C2×S3×D5)⋊1C22, (D5×C2×C6)⋊6C22, (C5×C4○D12)⋊3C2, C22.10(C2×S3×D5), C2.24(C22×S3×D5), (C5×C3⋊D4)⋊10C22, (C2×C10).12(C22×S3), (C2×C6).248(C22×D5), SmallGroup(480,1093)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — D20⋊25D6
Chief series
C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — D5×C3⋊D4 — D20⋊25D6
Lower central
C15 — C30 — D20⋊25D6

Subgroups: 1916 in 332 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], C5, S3 [×4], C6, C6 [×5], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], D5 [×6], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×6], C15, C2×D4 [×9], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], D10 [×8], C2×C10, C2×C10 [×2], Dic6, Dic6, C4×S3 [×2], C4×S3 [×2], D12, D12, C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×10], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×2], C5×S3 [×2], C3×D5 [×4], D15 [×2], C30, C30, 2+ (1+4), Dic10, C4×D5 [×6], D20 [×4], D20 [×5], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5 [×2], C22×D5 [×4], C4○D12, C4○D12, S3×D4 [×4], D4⋊2S3 [×4], C2×C3⋊D4 [×4], C6×D4, C5×Dic3 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×D20, C2×D20 [×2], C4○D20 [×3], D4×D5 [×6], Q8⋊2D5 [×2], C5×C4○D4, D4⋊6D6, D5×Dic3 [×4], C15⋊D4 [×4], C3⋊D20 [×4], C3×D20 [×4], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15 [×2], D60, C15⋊7D4 [×2], C2×C60, C2×S3×D5 [×4], D5×C2×C6 [×2], D4⋊8D10, D20⋊5S3 [×2], D20⋊S3 [×2], S3×D20 [×2], C20⋊D6 [×2], D5×C3⋊D4 [×4], C6×D20, C5×C4○D12, D60⋊11C2, D20⋊25D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ (1+4), C22×D5 [×7], S3×C23, S3×D5, C23×D5, D4⋊6D6, C2×S3×D5 [×3], D4⋊8D10, C22×S3×D5, D20⋊25D6

Generators and relations
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=cac-1=a-1, dad=a9, cbc-1=a8b, dbd=a18b, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽61)

0600000
1180000
000100
0060000
0000060
000010
,
0600000
6000000
000100
001000
0000060
0000600
,
100000
43600000
0004800
0048000
0000014
0000140
,
6000000
1810000
0000014
0000140
0004800
0048000

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,18,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,0,48,0,0,0,0,48,0,0,0,0,0,0,0,0,14,0,0,0,0,14,0],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,48,0,0,0,0,48,0,0,0,0,14,0,0,0,0,14,0,0,0] >;
 

63 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A10B10C10D10E10F10G10H12A12B15A15B20A20B20C20D20E20F20G20H20I20J30A···30F60A···60H
order1222222222234444445566666661010101010101010121215152020202020202020202030···3060···60
size1126610101010303022266303022222202020202244121212124444222244121212124···44···4

63 irreducible representations

dim11111111122222222224444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D5D6D6D6D10D10D10D10D102+ (1+4)S3×D5D4⋊6D6C2×S3×D5C2×S3×D5D4⋊8D10D20⋊25D6
kernelD20⋊25D6D20⋊5S3D20⋊S3S3×D20C20⋊D6D5×C3⋊D4C6×D20C5×C4○D12D60⋊11C2C2×D20C4○D12D20C2×C20C22×D5Dic6C4×S3D12C3⋊D4C2×C12C15C2×C4C5C4C22C3C1
# reps12222411112412242421224248

In GAP, Magma, Sage, TeX 

D_{20}\rtimes_{25}D_6
 
% in TeX
 
G:=Group("D20:25D6");
 
// GroupNames label
 
G:=SmallGroup(480,1093);
 
// by ID
 
G=gap.SmallGroup(480,1093);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,80,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^8*b,d*b*d=a^18*b,d*c*d=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀