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G = C3×D204C4order 480 = 25·3·5

Direct product of C3 and D204C4

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

Aliases: C3×D204C4, D204C12, C60.186D4, C12.69D20, Dic104C12, C1516C4≀C2, (C4×C20)⋊12C6, (C4×C60)⋊14C2, (C4×C12)⋊10D5, C4.6(D5×C12), C426(C3×D5), (C3×D20)⋊13C4, C4○D20.1C6, C20.33(C3×D4), C4.17(C3×D20), C12.63(C4×D5), C4.Dic51C6, C60.198(C2×C4), C20.37(C2×C12), (C2×C30).152D4, (C3×Dic10)⋊13C4, (C2×C12).420D10, C30.80(C22⋊C4), (C2×C60).516C22, C6.33(D10⋊C4), C53(C3×C4≀C2), (C2×C4).67(C6×D5), (C2×C20).99(C2×C6), (C3×C4○D20).7C2, (C2×C10).27(C3×D4), C22.7(C3×C5⋊D4), C2.3(C3×D10⋊C4), (C2×C6).60(C5⋊D4), C10.12(C3×C22⋊C4), (C3×C4.Dic5)⋊13C2, SmallGroup(480,83)

Series: Derived Chief Lower central Upper central

C1C20 — C3×D204C4
C1C5C10C20C2×C20C2×C60C3×C4○D20 — C3×D204C4
C5C10C20 — C3×D204C4
C1C12C2×C12C4×C12

Generators and relations for C3×D204C4
 G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b15c >

Subgroups: 288 in 88 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, C6, C6 [×2], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, D5, C10, C10, C12 [×2], C12 [×3], C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C24, C2×C12, C2×C12 [×2], C3×D4 [×2], C3×Q8, C3×D5, C30, C30, C4≀C2, C52C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5, C60 [×2], C60 [×2], C6×D5, C2×C30, C4.Dic5, C4×C20, C4○D20, C3×C4≀C2, C3×C52C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, C2×C60, D204C4, C3×C4.Dic5, C4×C60, C3×C4○D20, C3×D204C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D10, C2×C12, C3×D4 [×2], C3×D5, C4≀C2, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C4≀C2, D5×C12, C3×D20, C3×C5⋊D4, D204C4, C3×D10⋊C4, C3×D204C4

Smallest permutation representation of C3×D204C4
On 120 points
Generators in S120
(1 41 31)(2 42 32)(3 43 33)(4 44 34)(5 45 35)(6 46 36)(7 47 37)(8 48 38)(9 49 39)(10 50 40)(11 51 21)(12 52 22)(13 53 23)(14 54 24)(15 55 25)(16 56 26)(17 57 27)(18 58 28)(19 59 29)(20 60 30)(61 114 84)(62 115 85)(63 116 86)(64 117 87)(65 118 88)(66 119 89)(67 120 90)(68 101 91)(69 102 92)(70 103 93)(71 104 94)(72 105 95)(73 106 96)(74 107 97)(75 108 98)(76 109 99)(77 110 100)(78 111 81)(79 112 82)(80 113 83)
(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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(29 85)(30 84)(31 83)(32 82)(33 81)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 113)(42 112)(43 111)(44 110)(45 109)(46 108)(47 107)(48 106)(49 105)(50 104)(51 103)(52 102)(53 101)(54 120)(55 119)(56 118)(57 117)(58 116)(59 115)(60 114)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)

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

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

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

138 conjugacy classes

class 1 2A2B2C3A3B4A4B4C···4G4H5A5B6A6B6C6D6E6F8A8B10A···10F12A12B12C12D12E···12N12O12P15A15B15C15D20A···20X24A24B24C24D30A···30L60A···60AV
order122233444···44556666668810···101212121212···1212121515151520···202424242430···3060···60
size1122011112···220221122202020202···211112···2202022222···2202020202···22···2

138 irreducible representations

dim111111111111222222222222222222
type+++++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12D4D4D5D10C3×D4C3×D4C3×D5C4≀C2C4×D5D20C5⋊D4C6×D5C3×C4≀C2D5×C12C3×D20C3×C5⋊D4D204C4C3×D204C4
kernelC3×D204C4C3×C4.Dic5C4×C60C3×C4○D20D204C4C3×Dic10C3×D20C4.Dic5C4×C20C4○D20Dic10D20C60C2×C30C4×C12C2×C12C20C2×C10C42C15C12C12C2×C6C2×C4C5C4C4C22C3C1
# reps11112222224411222244444488881632

Matrix representation of C3×D204C4 in GL3(𝔽241) generated by

1500
010
001
,
100
0400
00235
,
24000
00235
0400
,
24000
02400
00177
G:=sub<GL(3,GF(241))| [15,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,235],[240,0,0,0,0,40,0,235,0],[240,0,0,0,240,0,0,0,177] >;

C3×D204C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_4C_4
% in TeX

G:=Group("C3xD20:4C4");
// GroupNames label

G:=SmallGroup(480,83);
// by ID

G=gap.SmallGroup(480,83);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,1683,2524,102,18822]);
// Polycyclic

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

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