direct product, metabelian, soluble, monomial, A-group
Aliases: C5×C32⋊2C8, C32⋊2C40, (C3×C15)⋊8C8, (C3×C6).C20, (C3×C30).5C4, C10.3(C32⋊C4), C3⋊Dic3.1C10, C2.(C5×C32⋊C4), (C5×C3⋊Dic3).3C2, SmallGroup(360,55)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C6 — C3⋊Dic3 — C5×C3⋊Dic3 — C5×C32⋊2C8 |
| C32 — C5×C32⋊2C8 |
Generators and relations for C5×C32⋊2C8
G = < a,b,c,d | a5=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Smallest permutation representation of C5×C32⋊2C8
►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)
(2 19 104)(4 98 21)(6 23 100)(8 102 17)(10 42 31)(12 25 44)(14 46 27)(16 29 48)(33 70 62)(35 64 72)(37 66 58)(39 60 68)(50 118 88)(52 82 120)(54 114 84)(56 86 116)(73 109 96)(75 90 111)(77 105 92)(79 94 107)
(1 18 103)(2 19 104)(3 97 20)(4 98 21)(5 22 99)(6 23 100)(7 101 24)(8 102 17)(9 41 30)(10 42 31)(11 32 43)(12 25 44)(13 45 26)(14 46 27)(15 28 47)(16 29 48)(33 70 62)(34 63 71)(35 64 72)(36 65 57)(37 66 58)(38 59 67)(39 60 68)(40 69 61)(49 117 87)(50 118 88)(51 81 119)(52 82 120)(53 113 83)(54 114 84)(55 85 115)(56 86 116)(73 109 96)(74 89 110)(75 90 111)(76 112 91)(77 105 92)(78 93 106)(79 94 107)(80 108 95)
(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), (2,19,104)(4,98,21)(6,23,100)(8,102,17)(10,42,31)(12,25,44)(14,46,27)(16,29,48)(33,70,62)(35,64,72)(37,66,58)(39,60,68)(50,118,88)(52,82,120)(54,114,84)(56,86,116)(73,109,96)(75,90,111)(77,105,92)(79,94,107), (1,18,103)(2,19,104)(3,97,20)(4,98,21)(5,22,99)(6,23,100)(7,101,24)(8,102,17)(9,41,30)(10,42,31)(11,32,43)(12,25,44)(13,45,26)(14,46,27)(15,28,47)(16,29,48)(33,70,62)(34,63,71)(35,64,72)(36,65,57)(37,66,58)(38,59,67)(39,60,68)(40,69,61)(49,117,87)(50,118,88)(51,81,119)(52,82,120)(53,113,83)(54,114,84)(55,85,115)(56,86,116)(73,109,96)(74,89,110)(75,90,111)(76,112,91)(77,105,92)(78,93,106)(79,94,107)(80,108,95), (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), (2,19,104)(4,98,21)(6,23,100)(8,102,17)(10,42,31)(12,25,44)(14,46,27)(16,29,48)(33,70,62)(35,64,72)(37,66,58)(39,60,68)(50,118,88)(52,82,120)(54,114,84)(56,86,116)(73,109,96)(75,90,111)(77,105,92)(79,94,107), (1,18,103)(2,19,104)(3,97,20)(4,98,21)(5,22,99)(6,23,100)(7,101,24)(8,102,17)(9,41,30)(10,42,31)(11,32,43)(12,25,44)(13,45,26)(14,46,27)(15,28,47)(16,29,48)(33,70,62)(34,63,71)(35,64,72)(36,65,57)(37,66,58)(38,59,67)(39,60,68)(40,69,61)(49,117,87)(50,118,88)(51,81,119)(52,82,120)(53,113,83)(54,114,84)(55,85,115)(56,86,116)(73,109,96)(74,89,110)(75,90,111)(76,112,91)(77,105,92)(78,93,106)(79,94,107)(80,108,95), (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)], [(2,19,104),(4,98,21),(6,23,100),(8,102,17),(10,42,31),(12,25,44),(14,46,27),(16,29,48),(33,70,62),(35,64,72),(37,66,58),(39,60,68),(50,118,88),(52,82,120),(54,114,84),(56,86,116),(73,109,96),(75,90,111),(77,105,92),(79,94,107)], [(1,18,103),(2,19,104),(3,97,20),(4,98,21),(5,22,99),(6,23,100),(7,101,24),(8,102,17),(9,41,30),(10,42,31),(11,32,43),(12,25,44),(13,45,26),(14,46,27),(15,28,47),(16,29,48),(33,70,62),(34,63,71),(35,64,72),(36,65,57),(37,66,58),(38,59,67),(39,60,68),(40,69,61),(49,117,87),(50,118,88),(51,81,119),(52,82,120),(53,113,83),(54,114,84),(55,85,115),(56,86,116),(73,109,96),(74,89,110),(75,90,111),(76,112,91),(77,105,92),(78,93,106),(79,94,107),(80,108,95)], [(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)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 20A | ··· | 20H | 30A | ··· | 30H | 40A | ··· | 40P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 |
| size | 1 | 1 | 4 | 4 | 9 | 9 | 1 | 1 | 1 | 1 | 4 | 4 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 9 | ··· | 9 | 4 | ··· | 4 | 9 | ··· | 9 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C20 | C40 | C32⋊C4 | C32⋊2C8 | C5×C32⋊C4 | C5×C32⋊2C8 |
| kernel | C5×C32⋊2C8 | C5×C3⋊Dic3 | C3×C30 | C32⋊2C8 | C3×C15 | C3⋊Dic3 | C3×C6 | C32 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 | 2 | 2 | 8 | 8 |
Matrix representation of C5×C32⋊2C8 ►in GL5(𝔽241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 87 | 0 | 0 | 0 |
| 0 | 0 | 87 | 0 | 0 |
| 0 | 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 0 | 87 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 26 | 5 | 240 | 240 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 1 | 240 | 0 | 0 |
| 0 | 234 | 229 | 0 | 1 |
| 0 | 19 | 234 | 240 | 240 |
| 233 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 1 |
| 0 | 26 | 5 | 239 | 240 |
| 0 | 204 | 70 | 229 | 7 |
| 0 | 105 | 27 | 229 | 7 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,87,0,0,0,0,0,87,0,0,0,0,0,87,0,0,0,0,0,87],[1,0,0,0,0,0,1,0,0,26,0,0,1,0,5,0,0,0,0,240,0,0,0,1,240],[1,0,0,0,0,0,0,1,234,19,0,240,240,229,234,0,0,0,0,240,0,0,0,1,240],[233,0,0,0,0,0,0,26,204,105,0,0,5,70,27,0,240,239,229,229,0,1,240,7,7] >;
C5×C32⋊2C8 in GAP, Magma, Sage, TeX
C_5\times C_3^2\rtimes_2C_8
% in TeX
G:=Group("C5xC3^2:2C8"); // GroupNames label
G:=SmallGroup(360,55);
// by ID
G=gap.SmallGroup(360,55);
# by ID
G:=PCGroup([6,-2,-5,-2,-2,-3,3,60,50,8404,256,11525,881]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
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