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## G = C5×C32⋊2C8order 360 = 23·32·5

### Direct product of C5 and C32⋊2C8

Aliases: C5×C322C8, C322C40, (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

 Derived series C1 — C32 — C5×C32⋊2C8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C5×C3⋊Dic3 — C5×C32⋊2C8
 Lower central C32 — C5×C32⋊2C8
 Upper central C1 — C10

Generators and relations for C5×C322C8
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×C322C8
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×C322C8 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×C322C8 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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