Copied to
clipboard

G = C3×C20.D4order 480 = 25·3·5

Direct product of C3 and C20.D4

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

Aliases: C3×C20.D4, C60.118D4, (C6×D4).7D5, C20.8(C3×D4), (D4×C10).2C6, (D4×C30).7C2, C4.Dic53C6, C23.(C3×Dic5), (C22×C30).3C4, (C2×C12).213D10, C12.92(C5⋊D4), C1511(C4.D4), (C22×C10).4C12, C22.2(C6×Dic5), (C22×C6).1Dic5, (C2×C60).280C22, C6.23(C23.D5), C30.111(C22⋊C4), (C2×C4).3(C6×D5), C53(C3×C4.D4), (C2×D4).2(C3×D5), C4.13(C3×C5⋊D4), (C2×C20).16(C2×C6), (C2×C30).185(C2×C4), (C2×C10).49(C2×C12), C2.4(C3×C23.D5), C10.25(C3×C22⋊C4), (C3×C4.Dic5)⋊15C2, (C2×C6).20(C2×Dic5), SmallGroup(480,111)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C20.D4
C1C5C10C2×C10C2×C20C2×C60C3×C4.Dic5 — C3×C20.D4
C5C10C2×C10 — C3×C20.D4
C1C6C2×C12C6×D4

Generators and relations for C3×C20.D4
 G = < a,b,c,d | a3=b20=1, c4=b10, d2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b9, dcd-1=b15c3 >

Subgroups: 224 in 92 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C24 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×3], C4.D4, C52C8 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C3×M4(2) [×2], C6×D4, C60 [×2], C2×C30, C2×C30 [×4], C4.Dic5 [×2], D4×C10, C3×C4.D4, C3×C52C8 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C20.D4, C3×C4.Dic5 [×2], D4×C30, C3×C20.D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, Dic5 [×2], D10, C2×C12, C3×D4 [×2], C3×D5, C4.D4, C2×Dic5, C5⋊D4 [×2], C3×C22⋊C4, C3×Dic5 [×2], C6×D5, C23.D5, C3×C4.D4, C6×Dic5, C3×C5⋊D4 [×2], C20.D4, C3×C23.D5, C3×C20.D4

Smallest permutation representation of C3×C20.D4
On 120 points
Generators in S120
(1 41 30)(2 42 31)(3 43 32)(4 44 33)(5 45 34)(6 46 35)(7 47 36)(8 48 37)(9 49 38)(10 50 39)(11 51 40)(12 52 21)(13 53 22)(14 54 23)(15 55 24)(16 56 25)(17 57 26)(18 58 27)(19 59 28)(20 60 29)(61 116 91)(62 117 92)(63 118 93)(64 119 94)(65 120 95)(66 101 96)(67 102 97)(68 103 98)(69 104 99)(70 105 100)(71 106 81)(72 107 82)(73 108 83)(74 109 84)(75 110 85)(76 111 86)(77 112 87)(78 113 88)(79 114 89)(80 115 90)
(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 76 6 71 11 66 16 61)(2 75 7 70 12 65 17 80)(3 74 8 69 13 64 18 79)(4 73 9 68 14 63 19 78)(5 72 10 67 15 62 20 77)(21 95 26 90 31 85 36 100)(22 94 27 89 32 84 37 99)(23 93 28 88 33 83 38 98)(24 92 29 87 34 82 39 97)(25 91 30 86 35 81 40 96)(41 111 46 106 51 101 56 116)(42 110 47 105 52 120 57 115)(43 109 48 104 53 119 58 114)(44 108 49 103 54 118 59 113)(45 107 50 102 55 117 60 112)
(1 66 6 71 11 76 16 61)(2 75 7 80 12 65 17 70)(3 64 8 69 13 74 18 79)(4 73 9 78 14 63 19 68)(5 62 10 67 15 72 20 77)(21 95 26 100 31 85 36 90)(22 84 27 89 32 94 37 99)(23 93 28 98 33 83 38 88)(24 82 29 87 34 92 39 97)(25 91 30 96 35 81 40 86)(41 101 46 106 51 111 56 116)(42 110 47 115 52 120 57 105)(43 119 48 104 53 109 58 114)(44 108 49 113 54 118 59 103)(45 117 50 102 55 107 60 112)

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

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

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

93 conjugacy classes

class 1 2A2B2C2D3A3B4A4B5A5B6A6B6C6D6E6F6G6H8A8B8C8D10A···10F10G···10N12A12B12C12D15A15B15C15D20A20B20C20D24A···24H30A···30L30M···30AB60A···60H
order1222233445566666666888810···1010···1012121212151515152020202024···2430···3030···3060···60
size1124411222211224444202020202···24···422222222444420···202···24···44···4

93 irreducible representations

dim1111111122222222224444
type++++++-+
imageC1C2C2C3C4C6C6C12D4D5D10Dic5C3×D4C3×D5C5⋊D4C6×D5C3×Dic5C3×C5⋊D4C4.D4C3×C4.D4C20.D4C3×C20.D4
kernelC3×C20.D4C3×C4.Dic5D4×C30C20.D4C22×C30C4.Dic5D4×C10C22×C10C60C6×D4C2×C12C22×C6C20C2×D4C12C2×C4C23C4C15C5C3C1
# reps12124428222444848161248

Matrix representation of C3×C20.D4 in GL4(𝔽241) generated by

15000
01500
00150
00015
,
09800
143000
00091
001500
,
0010
000240
0100
1000
,
0010
0001
0100
240000
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[0,143,0,0,98,0,0,0,0,0,0,150,0,0,91,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,240,0,0],[0,0,0,240,0,0,1,0,1,0,0,0,0,1,0,0] >;

C3×C20.D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}.D_4
% in TeX

G:=Group("C3xC20.D4");
// GroupNames label

G:=SmallGroup(480,111);
// by ID

G=gap.SmallGroup(480,111);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,850,136,2524,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=1,c^4=b^10,d^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

׿
×
𝔽