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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — (C2×C6)⋊8D20
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — C2×C3⋊D20 — (C2×C6)⋊8D20
 Lower central C15 — C2×C30 — (C2×C6)⋊8D20
 Upper central C1 — C22 — C23

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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 5A 5B 6A ··· 6G 6H ··· 6O 10A ··· 10F 10G 10H 10I 10J 15A 15B 20A ··· 20H 30A ··· 30N order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 5 5 6 ··· 6 6 ··· 6 10 ··· 10 10 10 10 10 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 2 2 10 10 10 10 60 2 12 12 60 2 2 2 ··· 2 10 ··· 10 2 ··· 2 4 4 4 4 4 4 12 ··· 12 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 C3⋊D4 C3⋊D4 D20 S3×D5 D4×D5 C3⋊D20 C2×S3×D5 D5×C3⋊D4 kernel (C2×C6)⋊8D20 D10⋊Dic3 C5×C6.D4 C2×C3⋊D20 C2×C15⋊7D4 D5×C22×C6 C23×D5 C6×D5 C2×C30 C6.D4 C22×D5 C22×C10 C2×Dic3 C22×C6 D10 C2×C10 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 1 2 1 1 1 4 2 2 2 1 4 2 8 4 8 2 4 4 2 8

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

 60 0 0 0 0 60 0 0 0 0 60 29 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 14 5 0 0 0 48
,
 7 32 0 0 29 2 0 0 0 0 55 3 0 0 8 6
,
 7 32 0 0 29 54 0 0 0 0 55 49 0 0 8 6
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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