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## G = C3×D4⋊8D10order 480 = 25·3·5

### Direct product of C3 and D4⋊8D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×D4⋊8D10
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — C3×D4×D5 — C3×D4⋊8D10
 Lower central C5 — C10 — C3×D4⋊8D10
 Upper central C1 — C6 — C3×C4○D4

Generators and relations for C3×D48D10
G = < a,b,c,d,e | a3=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ce=ec, ede=d-1 >

Subgroups: 1136 in 332 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C12, C12, C12, C2×C6, C2×C6, C15, C2×D4, C4○D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C3×D5, C30, C30, 2+ 1+4, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C6×D4, C3×C4○D4, C3×C4○D4, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C2×D20, C4○D20, D4×D5, Q82D5, C5×C4○D4, C3×2+ 1+4, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, C2×C60, D4×C15, Q8×C15, D5×C2×C6, D48D10, C6×D20, C3×C4○D20, C3×D4×D5, C3×Q82D5, C15×C4○D4, C3×D48D10
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C24, D10, C22×C6, C3×D5, 2+ 1+4, C22×D5, C23×C6, C6×D5, C23×D5, C3×2+ 1+4, D5×C2×C6, D48D10, D5×C22×C6, C3×D48D10

Smallest permutation representation of C3×D48D10
On 120 points
Generators in S120
(1 22 12)(2 23 13)(3 24 14)(4 25 15)(5 21 11)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 55 41)(32 51 42)(33 52 43)(34 53 44)(35 54 45)(36 60 47)(37 56 48)(38 57 49)(39 58 50)(40 59 46)(61 81 77)(62 82 78)(63 83 79)(64 84 80)(65 85 71)(66 86 72)(67 87 73)(68 88 74)(69 89 75)(70 90 76)(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 37 7 32)(2 38 8 33)(3 39 9 34)(4 40 10 35)(5 36 6 31)(11 47 16 41)(12 48 17 42)(13 49 18 43)(14 50 19 44)(15 46 20 45)(21 60 26 55)(22 56 27 51)(23 57 28 52)(24 58 29 53)(25 59 30 54)(61 97 66 92)(62 98 67 93)(63 99 68 94)(64 100 69 95)(65 91 70 96)(71 101 76 106)(72 102 77 107)(73 103 78 108)(74 104 79 109)(75 105 80 110)(81 117 86 112)(82 118 87 113)(83 119 88 114)(84 120 89 115)(85 111 90 116)
(1 92)(2 98)(3 94)(4 100)(5 96)(6 91)(7 97)(8 93)(9 99)(10 95)(11 106)(12 102)(13 108)(14 104)(15 110)(16 101)(17 107)(18 103)(19 109)(20 105)(21 116)(22 112)(23 118)(24 114)(25 120)(26 111)(27 117)(28 113)(29 119)(30 115)(31 65)(32 61)(33 67)(34 63)(35 69)(36 70)(37 66)(38 62)(39 68)(40 64)(41 71)(42 77)(43 73)(44 79)(45 75)(46 80)(47 76)(48 72)(49 78)(50 74)(51 81)(52 87)(53 83)(54 89)(55 85)(56 86)(57 82)(58 88)(59 84)(60 90)
(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 5)(2 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 22)(23 25)(26 27)(28 30)(31 37)(32 36)(33 40)(34 39)(35 38)(41 48)(42 47)(43 46)(44 50)(45 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)(73 80)(74 79)(75 78)(76 77)(81 90)(82 89)(83 88)(84 87)(85 86)(91 97)(92 96)(93 95)(98 100)(101 107)(102 106)(103 105)(108 110)(111 117)(112 116)(113 115)(118 120)

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

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

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

111 conjugacy classes

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

111 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D5 D10 D10 D10 C3×D5 C6×D5 C6×D5 C6×D5 2+ 1+4 C3×2+ 1+4 D4⋊8D10 C3×D4⋊8D10 kernel C3×D4⋊8D10 C6×D20 C3×C4○D20 C3×D4×D5 C3×Q8⋊2D5 C15×C4○D4 D4⋊8D10 C2×D20 C4○D20 D4×D5 Q8⋊2D5 C5×C4○D4 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C15 C5 C3 C1 # reps 1 3 3 6 2 1 2 6 6 12 4 2 2 6 6 2 4 12 12 4 1 2 4 8

Matrix representation of C3×D48D10 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 25 54 0 0 11 36 0 0 16 40 32 7 16 37 54 29
,
 36 7 60 39 50 25 40 21 45 21 29 54 45 24 7 32
,
 0 17 0 0 43 43 0 0 53 47 1 44 0 6 17 17
,
 1 0 0 0 42 60 0 0 53 47 1 44 47 0 0 60
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[25,11,16,16,54,36,40,37,0,0,32,54,0,0,7,29],[36,50,45,45,7,25,21,24,60,40,29,7,39,21,54,32],[0,43,53,0,17,43,47,6,0,0,1,17,0,0,44,17],[1,42,53,47,0,60,47,0,0,0,1,0,0,0,44,60] >;

C3×D48D10 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_8D_{10}
% in TeX

G:=Group("C3xD4:8D10");
// GroupNames label

G:=SmallGroup(480,1146);
// by ID

G=gap.SmallGroup(480,1146);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,555,1571,192,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^10=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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