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## G = C5×C23.6D6order 480 = 25·3·5

### Direct product of C5 and C23.6D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C23.6D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×C30 — C10×C3⋊D4 — C5×C23.6D6
 Lower central C3 — C6 — C2×C6 — C5×C23.6D6
 Upper central C1 — C10 — C22×C10 — C5×C22⋊C4

Generators and relations for C5×C23.6D6
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e6=b, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde5 >

Subgroups: 308 in 104 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C2×D4, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C23⋊C4, C2×C20, C5×D4, C22×C10, C22×C10, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C5×Dic3, C60, S3×C10, C2×C30, C2×C30, C5×C22⋊C4, C5×C22⋊C4, D4×C10, C23.6D6, C10×Dic3, C10×Dic3, C5×C3⋊D4, C2×C60, S3×C2×C10, C22×C30, C5×C23⋊C4, C5×C6.D4, C15×C22⋊C4, C10×C3⋊D4, C5×C23.6D6
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, D6, C22⋊C4, C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C23⋊C4, C2×C20, C5×D4, D6⋊C4, S3×C10, C5×C22⋊C4, C23.6D6, S3×C20, C5×D12, C5×C3⋊D4, C5×C23⋊C4, C5×D6⋊C4, C5×C23.6D6

Smallest permutation representation of C5×C23.6D6
On 120 points
Generators in S120
(1 53 35 109 99)(2 54 36 110 100)(3 55 25 111 101)(4 56 26 112 102)(5 57 27 113 103)(6 58 28 114 104)(7 59 29 115 105)(8 60 30 116 106)(9 49 31 117 107)(10 50 32 118 108)(11 51 33 119 97)(12 52 34 120 98)(13 62 96 42 80)(14 63 85 43 81)(15 64 86 44 82)(16 65 87 45 83)(17 66 88 46 84)(18 67 89 47 73)(19 68 90 48 74)(20 69 91 37 75)(21 70 92 38 76)(22 71 93 39 77)(23 72 94 40 78)(24 61 95 41 79)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
(1 19)(2 8)(3 21)(4 10)(5 23)(6 12)(7 13)(9 15)(11 17)(14 20)(16 22)(18 24)(25 92)(26 32)(27 94)(28 34)(29 96)(30 36)(31 86)(33 88)(35 90)(37 43)(38 111)(39 45)(40 113)(41 47)(42 115)(44 117)(46 119)(48 109)(49 64)(50 56)(51 66)(52 58)(53 68)(54 60)(55 70)(57 72)(59 62)(61 67)(63 69)(65 71)(73 79)(74 99)(75 81)(76 101)(77 83)(78 103)(80 105)(82 107)(84 97)(85 91)(87 93)(89 95)(98 104)(100 106)(102 108)(110 116)(112 118)(114 120)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 85)(37 116)(38 117)(39 118)(40 119)(41 120)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(73 104)(74 105)(75 106)(76 107)(77 108)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)
(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 9)(2 20 14 8)(3 19)(4 6 16 18)(7 15)(10 24 22 12)(11 23)(13 21)(25 90)(26 28 87 89)(29 86)(30 36 91 85)(31 35)(32 95 93 34)(33 94)(37 43 116 110)(38 42)(39 120 118 41)(40 119)(44 115)(45 47 112 114)(48 111)(49 53)(50 61 71 52)(51 72)(54 69 63 60)(55 68)(56 58 65 67)(59 64)(62 70)(73 102 104 83)(74 101)(75 81 106 100)(76 80)(77 98 108 79)(78 97)(82 105)(92 96)(99 107)(109 117)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A 10B 10C 10D 10E ··· 10P 10Q 10R 10S 10T 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I ··· 20T 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 3 4 4 4 4 4 5 5 5 5 6 6 6 6 6 10 10 10 10 10 ··· 10 10 10 10 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 2 2 2 12 2 4 4 12 12 12 1 1 1 1 2 2 2 4 4 1 1 1 1 2 ··· 2 12 12 12 12 4 4 4 4 2 2 2 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 S3 D4 D6 C4×S3 D12 C3⋊D4 C5×S3 C5×D4 S3×C10 S3×C20 C5×D12 C5×C3⋊D4 C23⋊C4 C23.6D6 C5×C23⋊C4 C5×C23.6D6 kernel C5×C23.6D6 C5×C6.D4 C15×C22⋊C4 C10×C3⋊D4 C10×Dic3 S3×C2×C10 C23.6D6 C6.D4 C3×C22⋊C4 C2×C3⋊D4 C2×Dic3 C22×S3 C5×C22⋊C4 C2×C30 C22×C10 C2×C10 C2×C10 C2×C10 C22⋊C4 C2×C6 C23 C22 C22 C22 C15 C5 C3 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 2 1 2 2 2 4 8 4 8 8 8 1 2 4 8

Matrix representation of C5×C23.6D6 in GL4(𝔽61) generated by

 58 0 0 0 0 58 0 0 0 0 58 0 0 0 0 58
,
 9 18 18 0 43 52 0 43 0 0 52 18 0 0 43 9
,
 52 43 0 43 18 9 18 0 0 0 52 18 0 0 43 9
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 9 52 20 20 9 52 21 20 0 27 9 9 34 0 52 52
,
 60 60 31 49 0 1 32 50 0 0 18 52 0 0 9 43
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[9,43,0,0,18,52,0,0,18,0,52,43,0,43,18,9],[52,18,0,0,43,9,0,0,0,18,52,43,43,0,18,9],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[9,9,0,34,52,52,27,0,20,21,9,52,20,20,9,52],[60,0,0,0,60,1,0,0,31,32,18,9,49,50,52,43] >;

C5×C23.6D6 in GAP, Magma, Sage, TeX

C_5\times C_2^3._6D_6
% in TeX

G:=Group("C5xC2^3.6D6");
// GroupNames label

G:=SmallGroup(480,125);
// by ID

G=gap.SmallGroup(480,125);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,2803,2111,15686]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^6=b,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;
// generators/relations

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