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## G = C3×D4⋊2Dic5order 480 = 25·3·5

### Direct product of C3 and D4⋊2Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D4⋊2Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C3×C4.Dic5 — C3×D4⋊2Dic5
 Lower central C5 — C10 — C20 — C3×D4⋊2Dic5
 Upper central C1 — C12 — C2×C12 — C3×C4○D4

Generators and relations for C3×D42Dic5
G = < a,b,c,d,e | a3=b4=d10=1, c2=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 224 in 88 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C4≀C2, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5, C60, C60, C2×C30, C2×C30, C4.Dic5, C4×Dic5, C5×C4○D4, C3×C4≀C2, C3×C52C8, C6×Dic5, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, D42Dic5, C3×C4.Dic5, C12×Dic5, C15×C4○D4, C3×D42Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, Dic5, D10, C2×C12, C3×D4, C3×D5, C4≀C2, C2×Dic5, C5⋊D4, C3×C22⋊C4, C3×Dic5, C6×D5, C23.D5, C3×C4≀C2, C6×Dic5, C3×C5⋊D4, D42Dic5, C3×C23.D5, C3×D42Dic5

Smallest permutation representation of C3×D42Dic5
On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(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)(61 101 81)(62 102 82)(63 103 83)(64 104 84)(65 105 85)(66 106 86)(67 107 87)(68 108 88)(69 109 89)(70 110 90)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)
(1 71 6 76)(2 77 7 72)(3 73 8 78)(4 79 9 74)(5 75 10 80)(11 66 16 61)(12 62 17 67)(13 68 18 63)(14 64 19 69)(15 70 20 65)(21 91 26 96)(22 97 27 92)(23 93 28 98)(24 99 29 94)(25 95 30 100)(31 86 36 81)(32 82 37 87)(33 88 38 83)(34 84 39 89)(35 90 40 85)(41 111 46 116)(42 117 47 112)(43 113 48 118)(44 119 49 114)(45 115 50 120)(51 106 56 101)(52 102 57 107)(53 108 58 103)(54 104 59 109)(55 110 60 105)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 71 66 76)(62 80 67 75)(63 79 68 74)(64 78 69 73)(65 77 70 72)(81 91 86 96)(82 100 87 95)(83 99 88 94)(84 98 89 93)(85 97 90 92)(101 111 106 116)(102 120 107 115)(103 119 108 114)(104 118 109 113)(105 117 110 112)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 6D 6E 6F 8A 8B 10A 10B 10C ··· 10H 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12P 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20J 24A 24B 24C 24D 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 8 8 10 10 10 ··· 10 12 12 12 12 12 12 12 12 12 ··· 12 15 15 15 15 20 20 20 20 20 ··· 20 24 24 24 24 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 4 1 1 1 1 2 4 10 10 10 10 2 2 1 1 2 2 4 4 20 20 2 2 4 ··· 4 1 1 1 1 2 2 4 4 10 ··· 10 2 2 2 2 2 2 2 2 4 ··· 4 20 20 20 20 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

102 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 2 2 2 2 2 2 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 D5 D10 Dic5 Dic5 C3×D4 C3×D4 C3×D5 C4≀C2 C5⋊D4 C5⋊D4 C6×D5 C3×Dic5 C3×Dic5 C3×C4≀C2 C3×C5⋊D4 C3×C5⋊D4 D4⋊2Dic5 C3×D4⋊2Dic5 kernel C3×D4⋊2Dic5 C3×C4.Dic5 C12×Dic5 C15×C4○D4 D4⋊2Dic5 D4×C15 Q8×C15 C4.Dic5 C4×Dic5 C5×C4○D4 C5×D4 C5×Q8 C60 C2×C30 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C20 C2×C10 C4○D4 C15 C12 C2×C6 C2×C4 D4 Q8 C5 C4 C22 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 4 8

Matrix representation of C3×D42Dic5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 64 0 0 0 0 0 177 177 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 64 128 0 0 0 0 0 177 0 0 0 0 0 0 1 0 0 0 0 0 1 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 240 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 1 51
,
 240 0 0 0 0 0 89 177 0 0 0 0 0 0 1 239 0 0 0 0 0 240 0 0 0 0 0 0 51 190 0 0 0 0 240 190

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,177,0,0,0,0,0,177,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,128,177,0,0,0,0,0,0,1,1,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,240,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,51],[240,89,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,239,240,0,0,0,0,0,0,51,240,0,0,0,0,190,190] >;

C3×D42Dic5 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_2{\rm Dic}_5
% in TeX

G:=Group("C3xD4:2Dic5");
// GroupNames label

G:=SmallGroup(480,115);
// by ID

G=gap.SmallGroup(480,115);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,136,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^4=d^10=1,c^2=b^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations

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