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## G = C3×Q8⋊2D5order 240 = 24·3·5

### Direct product of C3 and Q8⋊2D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×Q8⋊2D5
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C12 — C3×Q8⋊2D5
 Lower central C5 — C10 — C3×Q8⋊2D5
 Upper central C1 — C6 — C3×Q8

Generators and relations for C3×Q82D5
G = < a,b,c,d,e | a3=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 228 in 80 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, D4, Q8, D5, C10, C12, C12, C2×C6, C15, C4○D4, Dic5, C20, D10, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C4×D5, D20, C5×Q8, C3×C4○D4, C3×Dic5, C60, C6×D5, Q82D5, D5×C12, C3×D20, Q8×C15, C3×Q82D5
Quotients: C1, C2, C3, C22, C6, C23, D5, C2×C6, C4○D4, D10, C22×C6, C3×D5, C22×D5, C3×C4○D4, C6×D5, Q82D5, D5×C2×C6, C3×Q82D5

Smallest permutation representation of C3×Q82D5
On 120 points
Generators in S120
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(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 36 6 31)(2 37 7 32)(3 38 8 33)(4 39 9 34)(5 40 10 35)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)(61 91 66 96)(62 92 67 97)(63 93 68 98)(64 94 69 99)(65 95 70 100)(71 101 76 106)(72 102 77 107)(73 103 78 108)(74 104 79 109)(75 105 80 110)(81 111 86 116)(82 112 87 117)(83 113 88 118)(84 114 89 119)(85 115 90 120)
(1 66 6 61)(2 67 7 62)(3 68 8 63)(4 69 9 64)(5 70 10 65)(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 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 10)(7 9)(11 15)(12 14)(16 20)(17 19)(21 25)(22 24)(26 30)(27 29)(31 40)(32 39)(33 38)(34 37)(35 36)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 65)(62 64)(66 70)(67 69)(71 75)(72 74)(76 80)(77 79)(81 85)(82 84)(86 90)(87 89)(91 100)(92 99)(93 98)(94 97)(95 96)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)

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

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

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

C3×Q82D5 is a maximal subgroup of
D202Dic3  D20⋊D6  D20.13D6  D20.14D6  D20.D6  D20.Dic3  D20.29D6  D2016D6  D2017D6  C3×D5×C4○D4
C3×Q82D5 is a maximal quotient of
C3×Q8×Dic5

60 conjugacy classes

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

60 irreducible representations

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

Matrix representation of C3×Q82D5 in GL5(𝔽61)

 47 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 46 0 0 0 53 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 50 0 0 0 0 27 11
,
 1 0 0 0 0 0 0 60 0 0 0 1 43 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 18 60 0 0 0 18 43 0 0 0 0 0 1 0 0 0 0 53 60

G:=sub<GL(5,GF(61))| [47,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,53,0,0,0,46,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,50,27,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,60,43,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,18,18,0,0,0,60,43,0,0,0,0,0,1,53,0,0,0,0,60] >;

C3×Q82D5 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_2D_5
% in TeX

G:=Group("C3xQ8:2D5");
// GroupNames label

G:=SmallGroup(240,162);
// by ID

G=gap.SmallGroup(240,162);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,151,506,260,122,6917]);
// Polycyclic

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

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