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## G = C3×Q8⋊2F5order 480 = 25·3·5

### Direct product of C3 and Q8⋊2F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×Q8⋊2F5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C3×C4.F5 — C3×Q8⋊2F5
 Lower central C5 — C10 — C20 — C3×Q8⋊2F5
 Upper central C1 — C6 — C12 — C3×Q8

Generators and relations for C3×Q82F5
G = < a,b,c,d,e | a3=b4=d5=e4=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b-1c, ede-1=d3 >

Subgroups: 360 in 88 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, D4, Q8, D5, C10, C12, C12, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, C20, F5, D10, D10, C24, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C4≀C2, C5⋊C8, C4×D5, C4×D5, D20, D20, C5×Q8, C2×F5, C4×C12, C3×M4(2), C3×C4○D4, C3×Dic5, C60, C60, C3×F5, C6×D5, C6×D5, C4.F5, C4×F5, Q82D5, C3×C4≀C2, C3×C5⋊C8, D5×C12, D5×C12, C3×D20, C3×D20, Q8×C15, C6×F5, Q82F5, C3×C4.F5, C12×F5, C3×Q82D5, C3×Q82F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, F5, C2×C12, C3×D4, C4≀C2, C2×F5, C3×C22⋊C4, C3×F5, C22⋊F5, C3×C4≀C2, C6×F5, Q82F5, C3×C22⋊F5, C3×Q82F5

Smallest permutation representation of C3×Q82F5
On 120 points
Generators in S120
(1 44 24)(2 45 25)(3 41 21)(4 42 22)(5 43 23)(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 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(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 69 9 64)(2 70 10 65)(3 66 6 61)(4 67 7 62)(5 68 8 63)(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)
(2 3 5 4)(6 8 7 10)(11 13 12 15)(16 18 17 20)(21 23 22 25)(26 28 27 30)(31 33 32 35)(36 38 37 40)(41 43 42 45)(46 48 47 50)(51 53 52 55)(56 58 57 60)(61 78 67 75)(62 80 66 73)(63 77 70 71)(64 79 69 74)(65 76 68 72)(81 98 87 95)(82 100 86 93)(83 97 90 91)(84 99 89 94)(85 96 88 92)(101 118 107 115)(102 120 106 113)(103 117 110 111)(104 119 109 114)(105 116 108 112)

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D4 C3×D4 C3×D4 C4≀C2 C3×C4≀C2 F5 C2×F5 C3×F5 C22⋊F5 C6×F5 C3×C22⋊F5 Q8⋊2F5 C3×Q8⋊2F5 kernel C3×Q8⋊2F5 C3×C4.F5 C12×F5 C3×Q8⋊2D5 Q8⋊2F5 C3×D20 Q8×C15 C4.F5 C4×F5 Q8⋊2D5 D20 C5×Q8 C3×Dic5 C6×D5 Dic5 D10 C15 C5 C3×Q8 C12 Q8 C6 C4 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 2 2 4 8 1 1 2 2 2 4 1 2

Matrix representation of C3×Q82F5 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 15 0 0 0 0 0 0 15
,
 64 0 0 0 0 0 2 177 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
,
 79 5 0 0 0 0 53 162 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 240 240 240
,
 1 0 0 0 0 0 176 64 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 240 240 240 240

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[64,2,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[79,53,0,0,0,0,5,162,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[1,176,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;

C3×Q82F5 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes_2F_5
% in TeX

G:=Group("C3xQ8:2F5");
// GroupNames label

G:=SmallGroup(480,290);
// by ID

G=gap.SmallGroup(480,290);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,344,136,2524,1271,102,9414,1595]);
// Polycyclic

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

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