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G = D60⋊C4order 480 = 25·3·5

1st semidirect product of D60 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D601C4, D123F5, Dic5.1D12, C4⋊F51S3, C5⋊(C6.D8), C4.8(S3×F5), (C5×D12)⋊1C4, C20.1(C4×S3), C60.1(C2×C4), (C3×D5).1D8, C12.1(C2×F5), C32(D20⋊C4), (C4×D5).19D6, (C6×D5).17D4, (D5×D12).3C2, C2.4(D6⋊F5), C60.C41C2, C151(D4⋊C4), D5.2(D4⋊S3), C10.1(D6⋊C4), (C3×D5).1SD16, C6.1(C22⋊F5), C30.1(C22⋊C4), (C3×Dic5).20D4, D10.22(C3⋊D4), D5.2(Q82S3), (D5×C12).30C22, (C3×C4⋊F5)⋊1C2, SmallGroup(480,227)

Series: Derived Chief Lower central Upper central

C1C60 — D60⋊C4
C1C5C15C30C6×D5D5×C12C3×C4⋊F5 — D60⋊C4
C15C30C60 — D60⋊C4
C1C2C4

Generators and relations for D60⋊C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a43, cbc-1=a57b >

Subgroups: 788 in 100 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×5], C5, S3 [×2], C6, C6 [×2], C8, C2×C4 [×2], D4 [×3], C23, D5 [×2], D5, C10, C10, C12, C12 [×2], D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10 [×3], C2×C10, C3⋊C8, D12, D12 [×2], C2×C12 [×2], C22×S3, C5×S3, C3×D5 [×2], D15, C30, D4⋊C4, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C3×Dic5, C60, C3×F5, S3×D5 [×2], C6×D5, S3×C10, D30, D5⋊C8, C4⋊F5, D4×D5, C6.D8, C15⋊C8, C5⋊D12, D5×C12, C5×D12, D60, C6×F5, C2×S3×D5, D20⋊C4, C3×C4⋊F5, C60.C4, D5×D12, D60⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D8, SD16, F5, C4×S3, D12, C3⋊D4, D4⋊C4, C2×F5, D6⋊C4, D4⋊S3, Q82S3, C22⋊F5, C6.D8, S3×F5, D20⋊C4, D6⋊F5, D60⋊C4

Smallest permutation representation of D60⋊C4
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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 120)(10 119)(11 118)(12 117)(13 116)(14 115)(15 114)(16 113)(17 112)(18 111)(19 110)(20 109)(21 108)(22 107)(23 106)(24 105)(25 104)(26 103)(27 102)(28 101)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)
(2 8 50 44)(3 15 39 27)(4 22 28 10)(5 29 17 53)(6 36)(7 43 55 19)(9 57 33 45)(12 18 60 54)(13 25 49 37)(14 32 38 20)(16 46)(23 35 59 47)(24 42 48 30)(26 56)(34 52 58 40)(61 118 97 70)(62 65 86 113)(63 72 75 96)(64 79)(66 93 102 105)(67 100 91 88)(68 107 80 71)(69 114)(73 82 85 106)(74 89)(76 103 112 115)(77 110 101 98)(78 117 90 81)(83 92 95 116)(84 99)(87 120 111 108)(94 109)(104 119)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D 5 6A6B6C8A8B8C8D10A10B10C12A12B···12F 15  20  30 60A60B
order12222234444566688881010101212···121520306060
size11551260221020204210103030303042424420···2088888

33 irreducible representations

dim111111222222222444448888
type+++++++++++++++++++
imageC1C2C2C2C4C4S3D4D4D6D8SD16D12C4×S3C3⋊D4F5C2×F5D4⋊S3Q82S3C22⋊F5S3×F5D20⋊C4D6⋊F5D60⋊C4
kernelD60⋊C4C3×C4⋊F5C60.C4D5×D12C5×D12D60C4⋊F5C3×Dic5C6×D5C4×D5C3×D5C3×D5Dic5C20D10D12C12D5D5C6C4C3C2C1
# reps111122111122222111121112

Matrix representation of D60⋊C4 in GL8(𝔽241)

002402400000
00100000
11000000
2400000000
00000010
00000001
0000240240240240
00001000
,
11623212590000
1161251251160000
125912590000
1251161251160000
0000002400
0000024000
0000240000
00001111
,
10000000
01000000
0024000000
0002400000
00001000
00000001
00000100
0000240240240240

G:=sub<GL(8,GF(241))| [0,0,1,240,0,0,0,0,0,0,1,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,240,0],[116,116,125,125,0,0,0,0,232,125,9,116,0,0,0,0,125,125,125,125,0,0,0,0,9,116,9,116,0,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,240,0,1,0,0,0,0,240,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240] >;

D60⋊C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes C_4
% in TeX

G:=Group("D60:C4");
// GroupNames label

G:=SmallGroup(480,227);
// by ID

G=gap.SmallGroup(480,227);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,675,346,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^43,c*b*c^-1=a^57*b>;
// generators/relations

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