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G = C5×C3⋊C8order 120 = 23·3·5

Direct product of C5 and C3⋊C8

Aliases: C5×C3⋊C8, C3⋊C40, C155C8, C6.C20, C30.5C4, C60.6C2, C20.4S3, C12.2C10, C10.3Dic3, C4.2(C5×S3), C2.(C5×Dic3), SmallGroup(120,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C5×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C60 — C5×C3⋊C8
 Lower central C3 — C5×C3⋊C8
 Upper central C1 — C20

Generators and relations for C5×C3⋊C8
G = < a,b,c | a5=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×C3⋊C8
Regular action on 120 points
Generators in S120
(1 61 26 51 93)(2 62 27 52 94)(3 63 28 53 95)(4 64 29 54 96)(5 57 30 55 89)(6 58 31 56 90)(7 59 32 49 91)(8 60 25 50 92)(9 85 110 22 36)(10 86 111 23 37)(11 87 112 24 38)(12 88 105 17 39)(13 81 106 18 40)(14 82 107 19 33)(15 83 108 20 34)(16 84 109 21 35)(41 115 74 99 65)(42 116 75 100 66)(43 117 76 101 67)(44 118 77 102 68)(45 119 78 103 69)(46 120 79 104 70)(47 113 80 97 71)(48 114 73 98 72)
(1 18 103)(2 104 19)(3 20 97)(4 98 21)(5 22 99)(6 100 23)(7 24 101)(8 102 17)(9 41 30)(10 31 42)(11 43 32)(12 25 44)(13 45 26)(14 27 46)(15 47 28)(16 29 48)(33 62 70)(34 71 63)(35 64 72)(36 65 57)(37 58 66)(38 67 59)(39 60 68)(40 69 61)(49 87 117)(50 118 88)(51 81 119)(52 120 82)(53 83 113)(54 114 84)(55 85 115)(56 116 86)(73 109 96)(74 89 110)(75 111 90)(76 91 112)(77 105 92)(78 93 106)(79 107 94)(80 95 108)
(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)

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

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

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

C5×C3⋊C8 is a maximal subgroup of
D152C8  C20.32D6  D30.5C4  C3⋊D40  C6.D20  C15⋊SD16  C3⋊Dic20  S3×C40

60 conjugacy classes

 class 1 2 3 4A 4B 5A 5B 5C 5D 6 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 15A 15B 15C 15D 20A ··· 20H 30A 30B 30C 30D 40A ··· 40P 60A ··· 60H order 1 2 3 4 4 5 5 5 5 6 8 8 8 8 10 10 10 10 12 12 15 15 15 15 20 ··· 20 30 30 30 30 40 ··· 40 60 ··· 60 size 1 1 2 1 1 1 1 1 1 2 3 3 3 3 1 1 1 1 2 2 2 2 2 2 1 ··· 1 2 2 2 2 3 ··· 3 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C5 C8 C10 C20 C40 S3 Dic3 C3⋊C8 C5×S3 C5×Dic3 C5×C3⋊C8 kernel C5×C3⋊C8 C60 C30 C3⋊C8 C15 C12 C6 C3 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 4 4 8 16 1 1 2 4 4 8

Matrix representation of C5×C3⋊C8 in GL2(𝔽41) generated by

 16 0 0 16
,
 0 37 31 40
,
 3 29 0 38
G:=sub<GL(2,GF(41))| [16,0,0,16],[0,31,37,40],[3,0,29,38] >;

C5×C3⋊C8 in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes C_8
% in TeX

G:=Group("C5xC3:C8");
// GroupNames label

G:=SmallGroup(120,1);
// by ID

G=gap.SmallGroup(120,1);
# by ID

G:=PCGroup([5,-2,-5,-2,-2,-3,50,42,2004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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