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G = (C2×C12)⋊6F5order 480 = 25·3·5

4th semidirect product of C2×C12 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C60)⋊4C4, (C2×C12)⋊6F5, C60⋊C47C2, (D5×C12)⋊10C4, C60.60(C2×C4), (C4×D5)⋊5Dic3, (C4×D5).92D6, (C2×C20)⋊4Dic3, C12.53(C2×F5), (C6×Dic5)⋊15C4, C5⋊(C23.26D6), C154(C42⋊C2), C6.37(C22×F5), (C2×Dic5)⋊9Dic3, D5.5(C4○D12), C30.75(C22×C4), C20.21(C2×Dic3), (C6×D5).62C23, D10.17(C2×Dic3), D10.47(C22×S3), (C22×D5).103D6, C10.6(C22×Dic3), D10.D6.2C2, Dic5.17(C2×Dic3), (D5×C12).126C22, C34(D10.C23), (C4×C3⋊F5)⋊9C2, (C2×C4)⋊4(C3⋊F5), C4.21(C2×C3⋊F5), (C2×C4×D5).17S3, C2.7(C22×C3⋊F5), C22.7(C2×C3⋊F5), (D5×C2×C12).20C2, (C2×C6).48(C2×F5), (C2×C30).42(C2×C4), (C6×D5).60(C2×C4), (C2×C3⋊F5).15C22, (D5×C2×C6).145C22, (C3×D5).10(C4○D4), (C3×Dic5).67(C2×C4), (C2×C10).18(C2×Dic3), SmallGroup(480,1065)

Series: Derived Chief Lower central Upper central

C1C30 — (C2×C12)⋊6F5
C1C5C15C3×D5C6×D5C2×C3⋊F5C4×C3⋊F5 — (C2×C12)⋊6F5
C15C30 — (C2×C12)⋊6F5
C1C4C2×C4

Generators and relations for (C2×C12)⋊6F5
 G = < a,b,c,d | a2=b12=c5=d4=1, ab=ba, ac=ca, dad-1=ab6, bc=cb, dbd-1=b5, dcd-1=c3 >

Subgroups: 716 in 152 conjugacy classes, 61 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C42⋊C2, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×Dic3, C4⋊Dic3, C6.D4, C22×C12, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C23.26D6, D5×C12, C6×Dic5, C2×C60, C2×C3⋊F5, D5×C2×C6, D10.C23, C4×C3⋊F5, C60⋊C4, D10.D6, D5×C2×C12, (C2×C12)⋊6F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, F5, C2×Dic3, C22×S3, C42⋊C2, C2×F5, C4○D12, C22×Dic3, C3⋊F5, C22×F5, C23.26D6, C2×C3⋊F5, D10.C23, C22×C3⋊F5, (C2×C12)⋊6F5

Smallest permutation representation of (C2×C12)⋊6F5
On 120 points
Generators in S120
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 109)(13 106)(14 107)(15 108)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 85)(60 86)
(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 71 90 84 100)(2 72 91 73 101)(3 61 92 74 102)(4 62 93 75 103)(5 63 94 76 104)(6 64 95 77 105)(7 65 96 78 106)(8 66 85 79 107)(9 67 86 80 108)(10 68 87 81 97)(11 69 88 82 98)(12 70 89 83 99)(13 116 36 58 48)(14 117 25 59 37)(15 118 26 60 38)(16 119 27 49 39)(17 120 28 50 40)(18 109 29 51 41)(19 110 30 52 42)(20 111 31 53 43)(21 112 32 54 44)(22 113 33 55 45)(23 114 34 56 46)(24 115 35 57 47)
(2 6)(3 11)(5 9)(8 12)(13 42 36 52)(14 47 25 57)(15 40 26 50)(16 45 27 55)(17 38 28 60)(18 43 29 53)(19 48 30 58)(20 41 31 51)(21 46 32 56)(22 39 33 49)(23 44 34 54)(24 37 35 59)(61 88 102 82)(62 93 103 75)(63 86 104 80)(64 91 105 73)(65 96 106 78)(66 89 107 83)(67 94 108 76)(68 87 97 81)(69 92 98 74)(70 85 99 79)(71 90 100 84)(72 95 101 77)(109 111)(110 116)(112 114)(113 119)(115 117)(118 120)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G···4N 5 6A6B6C6D6E6F6G10A10B10C12A12B12C12D12E12F12G12H15A15B20A20B20C20D30A···30F60A···60H
order12222234444444···456666666101010121212121212121215152020202030···3060···60
size11255102112551030···304222101010104442222101010104444444···44···4

60 irreducible representations

dim111111112222222244444444
type++++++-+--++++
imageC1C2C2C2C2C4C4C4S3Dic3D6Dic3Dic3D6C4○D4C4○D12F5C2×F5C2×F5C3⋊F5C2×C3⋊F5C2×C3⋊F5D10.C23(C2×C12)⋊6F5
kernel(C2×C12)⋊6F5C4×C3⋊F5C60⋊C4D10.D6D5×C2×C12D5×C12C6×Dic5C2×C60C2×C4×D5C4×D5C4×D5C2×Dic5C2×C20C22×D5C3×D5D5C2×C12C12C2×C6C2×C4C4C22C3C1
# reps122214221221114812124248

Matrix representation of (C2×C12)⋊6F5 in GL4(𝔽61) generated by

701414
4754470
0475447
141407
,
8055
563560
056356
5508
,
60606060
1000
0100
0010
,
60000
00060
06000
1111
G:=sub<GL(4,GF(61))| [7,47,0,14,0,54,47,14,14,47,54,0,14,0,47,7],[8,56,0,5,0,3,56,5,5,56,3,0,5,0,56,8],[60,1,0,0,60,0,1,0,60,0,0,1,60,0,0,0],[60,0,0,1,0,0,60,1,0,0,0,1,0,60,0,1] >;

(C2×C12)⋊6F5 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes_6F_5
% in TeX

G:=Group("(C2xC12):6F5");
// GroupNames label

G:=SmallGroup(480,1065);
// by ID

G=gap.SmallGroup(480,1065);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,2693,14118,2379]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^6,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^3>;
// generators/relations

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