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G = D12⋊F5order 480 = 25·3·5

1st semidirect product of D12 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D604C4, D121F5, D5.2D24, D10.18D12, D5⋊C81S3, C5⋊(C2.D24), C4.1(S3×F5), (C5×D12)⋊4C4, C20.2(C4×S3), C60.8(C2×C4), C60⋊C41C2, (C3×D5).2D8, C31(D20⋊C4), (C6×D5).18D4, (C4×D5).20D6, (D5×D12).5C2, C2.5(D6⋊F5), C12.22(C2×F5), C152(D4⋊C4), C10.2(D6⋊C4), (C3×D5).2SD16, C6.2(C22⋊F5), D5.3(C24⋊C2), C30.2(C22⋊C4), (C3×Dic5).21D4, Dic5.1(C3⋊D4), (D5×C12).38C22, (C3×D5⋊C8)⋊1C2, SmallGroup(480,228)

Series: Derived Chief Lower central Upper central

C1C60 — D12⋊F5
C1C5C15C30C3×Dic5D5×C12C3×D5⋊C8 — D12⋊F5
C15C30C60 — D12⋊F5
C1C2C4

Generators and relations for D12⋊F5
 G = < a,b,c,d | a12=b2=c5=d4=1, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a7b, dcd-1=c3 >

Subgroups: 836 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, Dic3, C12, C12, D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10 [×3], C2×C10, C24, D12, D12 [×2], C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5 [×2], D15, C30, D4⋊C4, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C4⋊Dic3, C2×C24, C2×D12, C3×Dic5, C60, C3⋊F5, S3×D5 [×2], C6×D5, S3×C10, D30, D5⋊C8, C4⋊F5, D4×D5, C2.D24, C3×C5⋊C8, C5⋊D12, D5×C12, C5×D12, D60, C2×C3⋊F5, C2×S3×D5, D20⋊C4, C3×D5⋊C8, C60⋊C4, D5×D12, D12⋊F5
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, C24⋊C2, D24, D6⋊C4, C22⋊F5, C2.D24, S3×F5, D20⋊C4, D6⋊F5, D12⋊F5

Smallest permutation representation of D12⋊F5
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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 75)(14 74)(15 73)(16 84)(17 83)(18 82)(19 81)(20 80)(21 79)(22 78)(23 77)(24 76)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 72)(58 71)(59 70)(60 69)(97 109)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)
(1 52 96 74 111)(2 53 85 75 112)(3 54 86 76 113)(4 55 87 77 114)(5 56 88 78 115)(6 57 89 79 116)(7 58 90 80 117)(8 59 91 81 118)(9 60 92 82 119)(10 49 93 83 120)(11 50 94 84 109)(12 51 95 73 110)(13 106 41 64 30)(14 107 42 65 31)(15 108 43 66 32)(16 97 44 67 33)(17 98 45 68 34)(18 99 46 69 35)(19 100 47 70 36)(20 101 48 71 25)(21 102 37 72 26)(22 103 38 61 27)(23 104 39 62 28)(24 105 40 63 29)
(1 4)(2 3)(5 12)(6 11)(7 10)(8 9)(13 68 30 98)(14 67 31 97)(15 66 32 108)(16 65 33 107)(17 64 34 106)(18 63 35 105)(19 62 36 104)(20 61 25 103)(21 72 26 102)(22 71 27 101)(23 70 28 100)(24 69 29 99)(38 48)(39 47)(40 46)(41 45)(42 44)(49 90 120 80)(50 89 109 79)(51 88 110 78)(52 87 111 77)(53 86 112 76)(54 85 113 75)(55 96 114 74)(56 95 115 73)(57 94 116 84)(58 93 117 83)(59 92 118 82)(60 91 119 81)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D 5 6A6B6C8A8B8C8D10A10B10C12A12B12C12D 15  20 24A···24H 30 60A60B
order122222344445666888810101012121212152024···24306060
size115512602210606042101010101010424242210108810···10888

39 irreducible representations

dim111111222222222224448888
type++++++++++++++++++
imageC1C2C2C2C4C4S3D4D4D6D8SD16C3⋊D4C4×S3D12C24⋊C2D24F5C2×F5C22⋊F5S3×F5D20⋊C4D6⋊F5D12⋊F5
kernelD12⋊F5C3×D5⋊C8C60⋊C4D5×D12C5×D12D60D5⋊C8C3×Dic5C6×D5C4×D5C3×D5C3×D5Dic5C20D10D5D5D12C12C6C4C3C2C1
# reps111122111122222441121112

Matrix representation of D12⋊F5 in GL6(𝔽241)

142990000
142430000
001000
000100
000010
000001
,
1271360000
91140000
001000
000100
000010
000001
,
100000
010000
00240240240240
001000
000100
000010
,
1711010000
171700000
001000
000001
000100
00240240240240

G:=sub<GL(6,GF(241))| [142,142,0,0,0,0,99,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[127,9,0,0,0,0,136,114,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[171,171,0,0,0,0,101,70,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;

D12⋊F5 in GAP, Magma, Sage, TeX

D_{12}\rtimes F_5
% in TeX

G:=Group("D12:F5");
// GroupNames label

G:=SmallGroup(480,228);
// by ID

G=gap.SmallGroup(480,228);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,346,80,1356,9414,4724]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^5=d^4=1,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^3>;
// generators/relations

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