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G = D5×C2×C12order 240 = 24·3·5

Direct product of C2×C12 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5×C2×C12, C6012C22, C30.39C23, C308(C2×C4), C203(C2×C6), (C2×C20)⋊5C6, (C2×C60)⋊12C2, C102(C2×C12), C159(C22×C4), C52(C22×C12), Dic53(C2×C6), (C2×Dic5)⋊5C6, D10.8(C2×C6), (C2×C6).37D10, C22.9(C6×D5), (C6×Dic5)⋊11C2, C10.2(C22×C6), (C22×D5).5C6, C6.39(C22×D5), (C2×C30).38C22, (C6×D5).27C22, (C3×Dic5)⋊10C22, C2.1(D5×C2×C6), (D5×C2×C6).8C2, (C2×C10).9(C2×C6), SmallGroup(240,156)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C2×C12
C1C5C10C30C6×D5D5×C2×C6 — D5×C2×C12
C5 — D5×C2×C12
C1C2×C12

Generators and relations for D5×C2×C12
 G = < a,b,c,d | a2=b12=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 260 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C15, C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×4], C30, C30 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C22×C12, C3×Dic5 [×2], C60 [×2], C6×D5 [×6], C2×C30, C2×C4×D5, D5×C12 [×4], C6×Dic5, C2×C60, D5×C2×C6, D5×C2×C12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, D10 [×3], C2×C12 [×6], C22×C6, C3×D5, C4×D5 [×2], C22×D5, C22×C12, C6×D5 [×3], C2×C4×D5, D5×C12 [×2], D5×C2×C6, D5×C2×C12

Smallest permutation representation of D5×C2×C12
On 120 points
Generators in S120
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 94)(14 95)(15 96)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(37 71)(38 72)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 109)(58 110)(59 111)(60 112)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)
(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 47 21 49 83)(2 48 22 50 84)(3 37 23 51 73)(4 38 24 52 74)(5 39 13 53 75)(6 40 14 54 76)(7 41 15 55 77)(8 42 16 56 78)(9 43 17 57 79)(10 44 18 58 80)(11 45 19 59 81)(12 46 20 60 82)(25 67 88 111 105)(26 68 89 112 106)(27 69 90 113 107)(28 70 91 114 108)(29 71 92 115 97)(30 72 93 116 98)(31 61 94 117 99)(32 62 95 118 100)(33 63 96 119 101)(34 64 85 120 102)(35 65 86 109 103)(36 66 87 110 104)
(1 83)(2 84)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(25 105)(26 106)(27 107)(28 108)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 49)(48 50)(61 117)(62 118)(63 119)(64 120)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)

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

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

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

D5×C2×C12 is a maximal subgroup of
C60.93D4  C30.7M4(2)  D10.10D12  Dic3⋊C4⋊D5  D10⋊Dic6  (D5×C12)⋊C4  (C4×D5)⋊Dic3  C60.67D4  C60.68D4  D6⋊(C4×D5)  C1520(C4×D4)  D6⋊C4⋊D5  D10⋊D12  C60⋊D4  C127D20  C60.59(C2×C4)  (C2×C12)⋊6F5

96 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H5A5B6A···6F6G···6N10A···10F12A···12H12I···12P15A15B15C15D20A···20H30A···30L60A···60P
order122222223344444444556···66···610···1012···1212···121515151520···2030···3060···60
size111155551111115555221···15···52···21···15···522222···22···22···2

96 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C3C4C6C6C6C6C12D5D10D10C3×D5C4×D5C6×D5C6×D5D5×C12
kernelD5×C2×C12D5×C12C6×Dic5C2×C60D5×C2×C6C2×C4×D5C6×D5C4×D5C2×Dic5C2×C20C22×D5D10C2×C12C12C2×C6C2×C4C6C4C22C2
# reps1411128822216242488416

Matrix representation of D5×C2×C12 in GL3(𝔽61) generated by

6000
0600
0060
,
2100
0470
0047
,
100
001
06017
,
6000
001
010
G:=sub<GL(3,GF(61))| [60,0,0,0,60,0,0,0,60],[21,0,0,0,47,0,0,0,47],[1,0,0,0,0,60,0,1,17],[60,0,0,0,0,1,0,1,0] >;

D5×C2×C12 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{12}
% in TeX

G:=Group("D5xC2xC12");
// GroupNames label

G:=SmallGroup(240,156);
// by ID

G=gap.SmallGroup(240,156);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,122,6917]);
// Polycyclic

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

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