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## G = D5×C24⋊C2order 480 = 25·3·5

### Direct product of D5 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D5×C24⋊C2
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×D12 — D5×C24⋊C2
 Lower central C15 — C30 — C60 — D5×C24⋊C2
 Upper central C1 — C2 — C4 — C8

Generators and relations for D5×C24⋊C2
G = < a,b,c,d | a5=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >

Subgroups: 956 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, D5, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C24, C24, Dic6, Dic6, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×SD16, C52C8, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C24⋊C2, C24⋊C2, C2×C24, C2×Dic6, C2×D12, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, C2×C24⋊C2, C3×C52C8, C120, D5×Dic3, C5⋊D12, C15⋊Q8, D5×C12, C5×Dic6, C5×D12, Dic30, D60, C2×S3×D5, D5×SD16, D12.D5, Dic6⋊D5, D5×C24, C5×C24⋊C2, C24⋊D5, D5×Dic6, D5×D12, D5×C24⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, D12, C22×S3, C2×SD16, C22×D5, C24⋊C2, C2×D12, S3×D5, D4×D5, C2×C24⋊C2, C2×S3×D5, D5×SD16, D5×D12, D5×C24⋊C2

Smallest permutation representation of D5×C24⋊C2
On 120 points
Generators in S120
(1 59 82 120 43)(2 60 83 97 44)(3 61 84 98 45)(4 62 85 99 46)(5 63 86 100 47)(6 64 87 101 48)(7 65 88 102 25)(8 66 89 103 26)(9 67 90 104 27)(10 68 91 105 28)(11 69 92 106 29)(12 70 93 107 30)(13 71 94 108 31)(14 72 95 109 32)(15 49 96 110 33)(16 50 73 111 34)(17 51 74 112 35)(18 52 75 113 36)(19 53 76 114 37)(20 54 77 115 38)(21 55 78 116 39)(22 56 79 117 40)(23 57 80 118 41)(24 58 81 119 42)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 97)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 37)(26 48)(27 35)(28 46)(29 33)(30 44)(32 42)(34 40)(36 38)(39 47)(41 45)(49 69)(50 56)(51 67)(52 54)(53 65)(55 63)(57 61)(58 72)(60 70)(62 68)(64 66)(73 79)(74 90)(75 77)(76 88)(78 86)(80 84)(81 95)(83 93)(85 91)(87 89)(92 96)(97 107)(98 118)(99 105)(100 116)(101 103)(102 114)(104 112)(106 110)(109 119)(111 117)(113 115)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A 10B 10C 10D 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 8 8 8 8 10 10 10 10 12 12 12 12 15 15 20 20 20 20 24 24 24 24 24 24 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 5 5 12 60 2 2 10 12 60 2 2 2 10 10 2 2 10 10 2 2 24 24 2 2 10 10 4 4 4 4 24 24 2 2 2 2 10 10 10 10 4 4 4 4 4 4 4 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 SD16 D10 D10 D10 D12 D12 C24⋊C2 S3×D5 D4×D5 C2×S3×D5 D5×SD16 D5×D12 D5×C24⋊C2 kernel D5×C24⋊C2 D12.D5 Dic6⋊D5 D5×C24 C5×C24⋊C2 C24⋊D5 D5×Dic6 D5×D12 C8×D5 C3×Dic5 C6×D5 C24⋊C2 C5⋊2C8 C40 C4×D5 C3×D5 C24 Dic6 D12 Dic5 D10 D5 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 2 2 8 2 2 2 4 4 8

Matrix representation of D5×C24⋊C2 in GL6(𝔽241)

 1 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 0 189 1 0 0 0 0 240 0
,
 1 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 1 189 0 0 0 0 0 240
,
 203 73 0 0 0 0 208 0 0 0 0 0 0 0 240 1 0 0 0 0 240 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 123 0 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,240,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,189,240],[203,208,0,0,0,0,73,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,123,240,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D5×C24⋊C2 in GAP, Magma, Sage, TeX

D_5\times C_{24}\rtimes C_2
% in TeX

G:=Group("D5xC24:C2");
// GroupNames label

G:=SmallGroup(480,323);
// by ID

G=gap.SmallGroup(480,323);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,58,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^24=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations

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