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G = (C2×C6)⋊8D20order 480 = 25·3·5

2nd semidirect product of C2×C6 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C30)⋊5D4, (C2×C6)⋊8D20, C155C22≀C2, (C6×D5)⋊17D4, C6.162(D4×D5), C6.69(C2×D20), (C23×D5)⋊5S3, D108(C3⋊D4), C35(C22⋊D20), C51(C244S3), (C2×Dic3)⋊4D10, C30.244(C2×D4), C23.44(S3×D5), C6.D413D5, C223(C3⋊D20), (C22×D5).95D6, (C22×C10).58D6, (C22×C6).94D10, D10⋊Dic334C2, (C2×C30).206C23, (C10×Dic3)⋊8C22, (C22×D15)⋊6C22, (C2×Dic15)⋊12C22, (C22×C30).68C22, (D5×C22×C6)⋊2C2, C2.42(D5×C3⋊D4), (C2×C3⋊D20)⋊13C2, (C2×C157D4)⋊16C2, (C2×C10)⋊6(C3⋊D4), C10.66(C2×C3⋊D4), C2.24(C2×C3⋊D20), C22.235(C2×S3×D5), (D5×C2×C6).113C22, (C5×C6.D4)⋊15C2, (C2×C6).218(C22×D5), (C2×C10).218(C22×S3), SmallGroup(480,640)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C2×C6)⋊8D20
C1C5C15C30C2×C30D5×C2×C6C2×C3⋊D20 — (C2×C6)⋊8D20
C15C2×C30 — (C2×C6)⋊8D20
C1C22C23

Generators and relations for (C2×C6)⋊8D20
 G = < a,b,c,d | a2=b6=c20=d2=1, ab=ba, cac-1=dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1500 in 260 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3, C6, C6 [×2], C6 [×6], C2×C4 [×3], D4 [×6], C23, C23 [×9], D5 [×5], C10, C10 [×2], C10 [×2], Dic3 [×3], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×18], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×4], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×6], C22×S3, C22×C6, C22×C6 [×8], C3×D5 [×4], D15, C30, C30 [×2], C30 [×2], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×7], C22×C10, C6.D4, C6.D4 [×2], C2×C3⋊D4 [×3], C23×C6, C5×Dic3 [×2], Dic15, C6×D5 [×4], C6×D5 [×12], D30 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, C244S3, C3⋊D20 [×4], C10×Dic3 [×2], C2×Dic15, C157D4 [×2], D5×C2×C6 [×2], D5×C2×C6 [×6], C22×D15, C22×C30, C22⋊D20, D10⋊Dic3 [×2], C5×C6.D4, C2×C3⋊D20 [×2], C2×C157D4, D5×C22×C6, (C2×C6)⋊8D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, D20 [×2], C22×D5, C2×C3⋊D4 [×3], S3×D5, C2×D20, D4×D5 [×2], C244S3, C3⋊D20 [×2], C2×S3×D5, C22⋊D20, C2×C3⋊D20, D5×C3⋊D4 [×2], (C2×C6)⋊8D20

Smallest permutation representation of (C2×C6)⋊8D20
On 120 points
Generators in S120
(1 11)(2 30)(3 13)(4 32)(5 15)(6 34)(7 17)(8 36)(9 19)(10 38)(12 40)(14 22)(16 24)(18 26)(20 28)(21 31)(23 33)(25 35)(27 37)(29 39)(41 93)(42 52)(43 95)(44 54)(45 97)(46 56)(47 99)(48 58)(49 81)(50 60)(51 83)(53 85)(55 87)(57 89)(59 91)(61 71)(62 116)(63 73)(64 118)(65 75)(66 120)(67 77)(68 102)(69 79)(70 104)(72 106)(74 108)(76 110)(78 112)(80 114)(82 92)(84 94)(86 96)(88 98)(90 100)(101 111)(103 113)(105 115)(107 117)(109 119)
(1 63 58 39 107 100)(2 81 108 40 59 64)(3 65 60 21 109 82)(4 83 110 22 41 66)(5 67 42 23 111 84)(6 85 112 24 43 68)(7 69 44 25 113 86)(8 87 114 26 45 70)(9 71 46 27 115 88)(10 89 116 28 47 72)(11 73 48 29 117 90)(12 91 118 30 49 74)(13 75 50 31 119 92)(14 93 120 32 51 76)(15 77 52 33 101 94)(16 95 102 34 53 78)(17 79 54 35 103 96)(18 97 104 36 55 80)(19 61 56 37 105 98)(20 99 106 38 57 62)
(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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 40)(20 39)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 100)(58 99)(59 98)(60 97)(61 108)(62 107)(63 106)(64 105)(65 104)(66 103)(67 102)(68 101)(69 120)(70 119)(71 118)(72 117)(73 116)(74 115)(75 114)(76 113)(77 112)(78 111)(79 110)(80 109)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C5A5B6A···6G6H···6O10A···10F10G10H10I10J15A15B20A···20H30A···30N
order122222222223444556···66···610···1010101010151520···2030···30
size11112210101010602121260222···210···102···244444412···124···4

66 irreducible representations

dim1111112222222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10C3⋊D4C3⋊D4D20S3×D5D4×D5C3⋊D20C2×S3×D5D5×C3⋊D4
kernel(C2×C6)⋊8D20D10⋊Dic3C5×C6.D4C2×C3⋊D20C2×C157D4D5×C22×C6C23×D5C6×D5C2×C30C6.D4C22×D5C22×C10C2×Dic3C22×C6D10C2×C10C2×C6C23C6C22C22C2
# reps1212111422214284824428

Matrix representation of (C2×C6)⋊8D20 in GL4(𝔽61) generated by

60000
06000
006029
0001
,
1000
0100
00145
00048
,
73200
29200
00553
0086
,
73200
295400
005549
0086
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,29,1],[1,0,0,0,0,1,0,0,0,0,14,0,0,0,5,48],[7,29,0,0,32,2,0,0,0,0,55,8,0,0,3,6],[7,29,0,0,32,54,0,0,0,0,55,8,0,0,49,6] >;

(C2×C6)⋊8D20 in GAP, Magma, Sage, TeX

(C_2\times C_6)\rtimes_8D_{20}
% in TeX

G:=Group("(C2xC6):8D20");
// GroupNames label

G:=SmallGroup(480,640);
// by ID

G=gap.SmallGroup(480,640);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^20=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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