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## G = C3×D10.C23order 480 = 25·3·5

### Direct product of C3 and D10.C23

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D10.C23
 Chief series C1 — C5 — C10 — D10 — C6×D5 — C6×F5 — C12×F5 — C3×D10.C23
 Lower central C5 — C10 — C3×D10.C23
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×D10.C23
G = < a,b,c,d,e,f | a3=b10=c2=f2=1, d2=b-1c, e2=b5, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, dbd-1=b3, be=eb, bf=fb, dcd-1=b2c, ce=ec, cf=fc, de=ed, fdf=b5d, ef=fe >

Subgroups: 488 in 152 conjugacy classes, 72 normal (48 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C42⋊C2, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C3×Dic5, C60, C3×F5, C6×D5, C6×D5, C2×C30, C4×F5, C4⋊F5, C22⋊F5, C2×C4×D5, C3×C42⋊C2, D5×C12, C6×Dic5, C2×C60, C6×F5, D5×C2×C6, D10.C23, C12×F5, C3×C4⋊F5, C3×C22⋊F5, D5×C2×C12, C3×D10.C23
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C4○D4, F5, C2×C12, C22×C6, C42⋊C2, C2×F5, C22×C12, C3×C4○D4, C3×F5, C22×F5, C3×C42⋊C2, C6×F5, D10.C23, C2×C6×F5, C3×D10.C23

Smallest permutation representation of C3×D10.C23
On 120 points
Generators in S120
(1 27 17)(2 28 18)(3 29 19)(4 30 20)(5 21 11)(6 22 12)(7 23 13)(8 24 14)(9 25 15)(10 26 16)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 12)(13 20)(14 19)(15 18)(16 17)(21 22)(23 30)(24 29)(25 28)(26 27)(31 32)(33 40)(34 39)(35 38)(36 37)(41 42)(43 50)(44 49)(45 48)(46 47)(51 52)(53 60)(54 59)(55 58)(56 57)(61 62)(63 70)(64 69)(65 68)(66 67)(71 72)(73 80)(74 79)(75 78)(76 77)(81 82)(83 90)(84 89)(85 88)(86 87)(91 92)(93 100)(94 99)(95 98)(96 97)(101 102)(103 110)(104 109)(105 108)(106 107)(111 112)(113 120)(114 119)(115 118)(116 117)
(2 8 10 4)(3 5 9 7)(11 15 13 19)(14 16 20 18)(21 25 23 29)(24 26 30 28)(31 40 33 34)(32 37)(35 38 39 36)(41 50 43 44)(42 47)(45 48 49 46)(51 60 53 54)(52 57)(55 58 59 56)(61 65 63 69)(64 66 70 68)(71 75 73 79)(74 76 80 78)(81 85 83 89)(84 86 90 88)(91 100 93 94)(92 97)(95 98 99 96)(101 110 103 104)(102 107)(105 108 109 106)(111 120 113 114)(112 117)(115 118 119 116)
(1 62 6 67)(2 63 7 68)(3 64 8 69)(4 65 9 70)(5 66 10 61)(11 76 16 71)(12 77 17 72)(13 78 18 73)(14 79 19 74)(15 80 20 75)(21 86 26 81)(22 87 27 82)(23 88 28 83)(24 89 29 84)(25 90 30 85)(31 96 36 91)(32 97 37 92)(33 98 38 93)(34 99 39 94)(35 100 40 95)(41 106 46 101)(42 107 47 102)(43 108 48 103)(44 109 49 104)(45 110 50 105)(51 116 56 111)(52 117 57 112)(53 118 58 113)(54 119 59 114)(55 120 60 115)
(1 37)(2 38)(3 39)(4 40)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C12 C12 C12 C4○D4 C3×C4○D4 F5 C2×F5 C2×F5 C3×F5 C6×F5 C6×F5 D10.C23 C3×D10.C23 kernel C3×D10.C23 C12×F5 C3×C4⋊F5 C3×C22⋊F5 D5×C2×C12 D10.C23 D5×C12 C6×Dic5 C2×C60 C4×F5 C4⋊F5 C22⋊F5 C2×C4×D5 C4×D5 C2×Dic5 C2×C20 C3×D5 D5 C2×C12 C12 C2×C6 C2×C4 C4 C22 C3 C1 # reps 1 2 2 2 1 2 4 2 2 4 4 4 2 8 4 4 4 8 1 2 1 2 4 2 4 8

Matrix representation of C3×D10.C23 in GL6(𝔽61)

 13 0 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 60 0 0 0 0 1 0 60 0 0 0 1 0 0 0 0 60 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 60 1 0 0 0 60 0 1 0 0 60 0 0 1 0 0 0 0 0 1
,
 50 0 0 0 0 0 50 11 0 0 0 0 0 0 1 0 60 0 0 0 0 0 60 1 0 0 0 1 60 0 0 0 0 0 60 0
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11
,
 1 59 0 0 0 0 0 60 0 0 0 0 0 0 7 47 0 14 0 0 0 54 47 14 0 0 14 47 54 0 0 0 14 0 47 7

G:=sub<GL(6,GF(61))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,1,1,1,1,0,0,60,0,0,0,0,0,0,60,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,1,1,1,1],[50,50,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60,0,0,0,1,0,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[1,0,0,0,0,0,59,60,0,0,0,0,0,0,7,0,14,14,0,0,47,54,47,0,0,0,0,47,54,47,0,0,14,14,0,7] >;

C3×D10.C23 in GAP, Magma, Sage, TeX

C_3\times D_{10}.C_2^3
% in TeX

G:=Group("C3xD10.C2^3");
// GroupNames label

G:=SmallGroup(480,1052);
// by ID

G=gap.SmallGroup(480,1052);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,344,1094,9414,818]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^10=c^2=f^2=1,d^2=b^-1*c,e^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,d*b*d^-1=b^3,b*e=e*b,b*f=f*b,d*c*d^-1=b^2*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=b^5*d,e*f=f*e>;
// generators/relations

׿
×
𝔽