Copied to
clipboard

G = C3×C22⋊D20order 480 = 25·3·5

Direct product of C3 and C22⋊D20

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

Aliases: C3×C22⋊D20, (C2×C6)⋊7D20, (C6×D5)⋊19D4, D104(C3×D4), (C2×D20)⋊2C6, (C2×C30)⋊16D4, C2.7(C6×D20), C1512C22≀C2, (C6×D20)⋊18C2, (C2×C12)⋊19D10, C10.17(C6×D4), C6.171(D4×D5), C6.76(C2×D20), (C23×D5)⋊4C6, C223(C3×D20), D10⋊C44C6, (C2×C60)⋊20C22, C30.278(C2×D4), C23.19(C6×D5), (C22×C6).75D10, (C2×C30).340C23, (C6×Dic5)⋊18C22, (C22×C30).98C22, C2.7(C3×D4×D5), (C2×C4)⋊1(C6×D5), (C2×C20)⋊1(C2×C6), C51(C3×C22≀C2), (C2×C10)⋊3(C3×D4), (C2×C5⋊D4)⋊1C6, (D5×C22×C6)⋊7C2, (C6×C5⋊D4)⋊16C2, (C5×C22⋊C4)⋊3C6, C22⋊C42(C3×D5), (D5×C2×C6)⋊13C22, C22.41(D5×C2×C6), (C3×C22⋊C4)⋊10D5, (C2×Dic5)⋊1(C2×C6), (C22×D5)⋊1(C2×C6), (C15×C22⋊C4)⋊12C2, (C3×D10⋊C4)⋊15C2, (C22×C10).17(C2×C6), (C2×C10).23(C22×C6), (C2×C6).336(C22×D5), SmallGroup(480,675)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C22⋊D20
C1C5C10C2×C10C2×C30D5×C2×C6D5×C22×C6 — C3×C22⋊D20
C5C2×C10 — C3×C22⋊D20
C1C2×C6C3×C22⋊C4

Generators and relations for C3×C22⋊D20
 G = < a,b,c,d,e | a3=b2=c2=d20=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1088 in 260 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, C6, C6 [×2], C6 [×7], C2×C4 [×2], C2×C4, D4 [×6], C23, C23 [×9], D5 [×5], C10, C10 [×2], C10 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×21], C15, C22⋊C4, C22⋊C4 [×2], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×4], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12, C3×D4 [×6], C22×C6, C22×C6 [×9], C3×D5 [×5], C30, C30 [×2], C30 [×2], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C22×D5, C22×D5 [×2], C22×D5 [×6], C22×C10, C3×C22⋊C4, C3×C22⋊C4 [×2], C6×D4 [×3], C23×C6, C3×Dic5, C60 [×2], C6×D5 [×4], C6×D5 [×15], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, C3×C22≀C2, C3×D20 [×4], C6×Dic5, C3×C5⋊D4 [×2], C2×C60 [×2], D5×C2×C6, D5×C2×C6 [×2], D5×C2×C6 [×6], C22×C30, C22⋊D20, C3×D10⋊C4 [×2], C15×C22⋊C4, C6×D20 [×2], C6×C5⋊D4, D5×C22×C6, C3×C22⋊D20
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, D5, C2×C6 [×7], C2×D4 [×3], D10 [×3], C3×D4 [×6], C22×C6, C3×D5, C22≀C2, D20 [×2], C22×D5, C6×D4 [×3], C6×D5 [×3], C2×D20, D4×D5 [×2], C3×C22≀C2, C3×D20 [×2], D5×C2×C6, C22⋊D20, C6×D20, C3×D4×D5 [×2], C3×C22⋊D20

Smallest permutation representation of C3×C22⋊D20
On 120 points
Generators in S120
(1 64 87)(2 65 88)(3 66 89)(4 67 90)(5 68 91)(6 69 92)(7 70 93)(8 71 94)(9 72 95)(10 73 96)(11 74 97)(12 75 98)(13 76 99)(14 77 100)(15 78 81)(16 79 82)(17 80 83)(18 61 84)(19 62 85)(20 63 86)(21 53 103)(22 54 104)(23 55 105)(24 56 106)(25 57 107)(26 58 108)(27 59 109)(28 60 110)(29 41 111)(30 42 112)(31 43 113)(32 44 114)(33 45 115)(34 46 116)(35 47 117)(36 48 118)(37 49 119)(38 50 120)(39 51 101)(40 52 102)
(1 11)(2 28)(3 13)(4 30)(5 15)(6 32)(7 17)(8 34)(9 19)(10 36)(12 38)(14 40)(16 22)(18 24)(20 26)(21 31)(23 33)(25 35)(27 37)(29 39)(41 51)(42 67)(43 53)(44 69)(45 55)(46 71)(47 57)(48 73)(49 59)(50 75)(52 77)(54 79)(56 61)(58 63)(60 65)(62 72)(64 74)(66 76)(68 78)(70 80)(81 91)(82 104)(83 93)(84 106)(85 95)(86 108)(87 97)(88 110)(89 99)(90 112)(92 114)(94 116)(96 118)(98 120)(100 102)(101 111)(103 113)(105 115)(107 117)(109 119)
(1 37)(2 38)(3 39)(4 40)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(41 76)(42 77)(43 78)(44 79)(45 80)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 40)(18 39)(19 38)(20 37)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(81 104)(82 103)(83 102)(84 101)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)

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

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

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

102 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B4A4B4C5A5B6A···6F6G6H6I6J6K···6R6S6T10A···10F10G10H10I10J12A12B12C12D12E12F15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order1222222222233444556···666666···66610···10101010101212121212121515151520···2030···3030···3060···60
size1111221010101020114420221···1222210···1020202···244444444202022224···42···24···44···4

102 irreducible representations

dim11111111111122222222222244
type+++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D5D10D10C3×D4C3×D4C3×D5D20C6×D5C6×D5C3×D20D4×D5C3×D4×D5
kernelC3×C22⋊D20C3×D10⋊C4C15×C22⋊C4C6×D20C6×C5⋊D4D5×C22×C6C22⋊D20D10⋊C4C5×C22⋊C4C2×D20C2×C5⋊D4C23×D5C6×D5C2×C30C3×C22⋊C4C2×C12C22×C6D10C2×C10C22⋊C4C2×C6C2×C4C23C22C6C2
# reps121211242422422428448841648

Matrix representation of C3×C22⋊D20 in GL4(𝔽61) generated by

13000
01300
00130
00013
,
60000
06000
00600
00551
,
1000
0100
00600
00060
,
22900
32700
00120
00660
,
22900
25900
006041
0001
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[60,0,0,0,0,60,0,0,0,0,60,55,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[2,32,0,0,29,7,0,0,0,0,1,6,0,0,20,60],[2,2,0,0,29,59,0,0,0,0,60,0,0,0,41,1] >;

C3×C22⋊D20 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes D_{20}
% in TeX

G:=Group("C3xC2^2:D20");
// GroupNames label

G:=SmallGroup(480,675);
// by ID

G=gap.SmallGroup(480,675);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,555,142,18822]);
// Polycyclic

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

׿
×
𝔽