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## G = (C2×C6).D20order 480 = 25·3·5

### 2nd non-split extension by C2×C6 of D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — (C2×C6).D20
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×C30 — C6×C5⋊D4 — (C2×C6).D20
 Lower central C15 — C30 — C2×C30 — (C2×C6).D20
 Upper central C1 — C2 — C23

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

Subgroups: 540 in 104 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C23⋊C4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C6.D4, C6.D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C2×C30, C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C23.7D6, C6×Dic5, C3×C5⋊D4, C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, C23.1D10, C5×C6.D4, C30.38D4, C6×C5⋊D4, (C2×C6).D20
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C23⋊C4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C23.7D6, D5×Dic3, C15⋊D4, C3⋊D20, C23.1D10, D10⋊Dic3, (C2×C6).D20

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

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

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

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

57 conjugacy classes

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

57 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 4 4 4 type + + + + + + + - - + + + + + - - + image C1 C2 C2 C2 C4 C4 S3 D4 D5 Dic3 Dic3 D6 D10 C3⋊D4 C4×D5 D20 C5⋊D4 C23⋊C4 S3×D5 C23.7D6 D5×Dic3 C15⋊D4 C3⋊D20 C23.1D10 (C2×C6).D20 kernel (C2×C6).D20 C5×C6.D4 C30.38D4 C6×C5⋊D4 C6×Dic5 D5×C2×C6 C2×C5⋊D4 C2×C30 C6.D4 C2×Dic5 C22×D5 C22×C10 C22×C6 C2×C10 C2×C6 C2×C6 C2×C6 C15 C23 C5 C22 C22 C22 C3 C1 # reps 1 1 1 1 2 2 1 2 2 1 1 1 2 4 4 4 4 1 2 2 2 2 2 4 8

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

 60 0 0 0 0 60 0 0 33 44 1 0 33 44 0 1
,
 25 45 0 0 49 37 0 0 14 53 53 16 41 28 45 9
,
 36 48 60 42 59 38 18 17 60 58 17 31 13 27 17 31
,
 47 44 0 0 8 14 0 0 41 25 0 1 41 25 1 0
G:=sub<GL(4,GF(61))| [60,0,33,33,0,60,44,44,0,0,1,0,0,0,0,1],[25,49,14,41,45,37,53,28,0,0,53,45,0,0,16,9],[36,59,60,13,48,38,58,27,60,18,17,17,42,17,31,31],[47,8,41,41,44,14,25,25,0,0,0,1,0,0,1,0] >;

(C2×C6).D20 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,71);
// by ID

G=gap.SmallGroup(480,71);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,346,1356,18822]);
// Polycyclic

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

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