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G = C3×C23⋊D10order 480 = 25·3·5

Direct product of C3 and C23⋊D10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3×C23⋊D10, (D4×C10)⋊8C6, (C6×D5)⋊20D4, (C6×D4)⋊10D5, D105(C3×D4), (C2×C30)⋊17D4, C231(C6×D5), C1515C22≀C2, (D4×C30)⋊22C2, (C2×C12)⋊20D10, C10.38(C6×D4), C6.192(D4×D5), (C23×D5)⋊5C6, (C22×C6)⋊1D10, (C2×C60)⋊35C22, C30.407(C2×D4), C23.D510C6, D10⋊C414C6, (C2×C30).369C23, (C22×C30)⋊12C22, (C6×Dic5)⋊19C22, (C2×C4)⋊2(C6×D5), C2.25(C3×D4×D5), (C2×C20)⋊7(C2×C6), C52(C3×C22≀C2), (C2×D4)⋊3(C3×D5), (C2×C10)⋊4(C3×D4), (C2×C5⋊D4)⋊4C6, (D5×C22×C6)⋊8C2, (C6×C5⋊D4)⋊19C2, C2.13(C6×C5⋊D4), C223(C3×C5⋊D4), (C2×C6)⋊10(C5⋊D4), C22.59(D5×C2×C6), (C22×C10)⋊4(C2×C6), (C2×Dic5)⋊2(C2×C6), C6.134(C2×C5⋊D4), (C3×C23.D5)⋊26C2, (C3×D10⋊C4)⋊36C2, (D5×C2×C6).135C22, (C2×C10).52(C22×C6), (C22×D5).30(C2×C6), (C2×C6).365(C22×D5), SmallGroup(480,730)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C23⋊D10
C1C5C10C2×C10C2×C30D5×C2×C6D5×C22×C6 — C3×C23⋊D10
C5C2×C10 — C3×C23⋊D10
C1C2×C6C6×D4

Generators and relations for C3×C23⋊D10
 G = < a,b,c,d,e,f | a3=b2=c2=d2=e10=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 992 in 260 conjugacy classes, 74 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, C6, C6 [×2], C6 [×7], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], D5 [×4], C10, C10 [×2], C10 [×3], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×21], C15, C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, Dic5 [×2], C20, D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C12, C2×C12 [×2], C3×D4 [×6], C22×C6 [×2], C22×C6 [×8], C3×D5 [×4], C30, C30 [×2], C30 [×3], C22≀C2, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10 [×2], C3×C22⋊C4 [×3], C6×D4, C6×D4 [×2], C23×C6, C3×Dic5 [×2], C60, C6×D5 [×4], C6×D5 [×12], C2×C30, C2×C30 [×2], C2×C30 [×5], D10⋊C4 [×2], C23.D5, C2×C5⋊D4 [×2], D4×C10, C23×D5, C3×C22≀C2, C6×Dic5 [×2], C3×C5⋊D4 [×4], C2×C60, D4×C15 [×2], D5×C2×C6 [×2], D5×C2×C6 [×6], C22×C30 [×2], C23⋊D10, C3×D10⋊C4 [×2], C3×C23.D5, C6×C5⋊D4 [×2], D4×C30, D5×C22×C6, C3×C23⋊D10
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, D5, C2×C6 [×7], C2×D4 [×3], D10 [×3], C3×D4 [×6], C22×C6, C3×D5, C22≀C2, C5⋊D4 [×2], C22×D5, C6×D4 [×3], C6×D5 [×3], D4×D5 [×2], C2×C5⋊D4, C3×C22≀C2, C3×C5⋊D4 [×2], D5×C2×C6, C23⋊D10, C3×D4×D5 [×2], C6×C5⋊D4, C3×C23⋊D10

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

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

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

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

102 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B4A4B4C5A5B6A···6F6G6H6I6J6K6L6M···6T10A···10F10G···10N12A12B12C12D12E12F15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order1222222222233444556···66666666···610···1010···10121212121212151515152020202030···3030···3060···60
size1111224101010101142020221···122224410···102···24···44420202020222244442···24···44···4

102 irreducible representations

dim11111111111122222222222244
type++++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D5D10D10C3×D4C3×D4C3×D5C5⋊D4C6×D5C6×D5C3×C5⋊D4D4×D5C3×D4×D5
kernelC3×C23⋊D10C3×D10⋊C4C3×C23.D5C6×C5⋊D4D4×C30D5×C22×C6C23⋊D10D10⋊C4C23.D5C2×C5⋊D4D4×C10C23×D5C6×D5C2×C30C6×D4C2×C12C22×C6D10C2×C10C2×D4C2×C6C2×C4C23C22C6C2
# reps121211242422422248448481648

Matrix representation of C3×C23⋊D10 in GL4(𝔽61) generated by

47000
04700
00130
00013
,
15300
06000
00311
001630
,
1000
0100
00600
00060
,
60000
06000
0010
0001
,
1000
466000
00018
004443
,
60000
15100
0010
006060
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,53,60,0,0,0,0,31,16,0,0,1,30],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[1,46,0,0,0,60,0,0,0,0,0,44,0,0,18,43],[60,15,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;

C3×C23⋊D10 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC2^3:D10");
// GroupNames label

G:=SmallGroup(480,730);
// by ID

G=gap.SmallGroup(480,730);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,555,18822]);
// Polycyclic

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

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