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## G = C3×Q8⋊F5order 480 = 25·3·5

### Direct product of C3 and Q8⋊F5

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×Q8⋊F5
G = < a,b,c,d,e | a3=b4=d5=e4=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d3 >

Subgroups: 312 in 84 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, Q8, Q8, D5, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C24, C2×C12, C3×Q8, C3×Q8, C3×D5, C30, Q8⋊C4, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C3×C4⋊C4, C2×C24, C6×Q8, C3×Dic5, C3×Dic5, C60, C60, C3×F5, C6×D5, D5⋊C8, C4⋊F5, Q8×D5, C3×Q8⋊C4, C3×C5⋊C8, C3×Dic10, C3×Dic10, D5×C12, D5×C12, Q8×C15, C6×F5, Q8⋊F5, C3×D5⋊C8, C3×C4⋊F5, C3×Q8×D5, C3×Q8⋊F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, SD16, Q16, F5, C2×C12, C3×D4, Q8⋊C4, C2×F5, C3×C22⋊C4, C3×SD16, C3×Q16, C3×F5, C22⋊F5, C3×Q8⋊C4, C6×F5, Q8⋊F5, C3×C22⋊F5, C3×Q8⋊F5

Smallest permutation representation of C3×Q8⋊F5
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 18 12 20)(13 17 15 16)(14 19)(21 23 22 25)(26 28 27 30)(31 38 32 40)(33 37 35 36)(34 39)(41 43 42 45)(46 48 47 50)(51 58 52 60)(53 57 55 56)(54 59)(61 73 62 75)(63 72 65 71)(64 74)(66 78 67 80)(68 77 70 76)(69 79)(81 93 82 95)(83 92 85 91)(84 94)(86 98 87 100)(88 97 90 96)(89 99)(101 113 102 115)(103 112 105 111)(104 114)(106 118 107 120)(108 117 110 116)(109 119)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 10 12A 12B 12C 12D 12E 12F 12G ··· 12L 15A 15B 20A 20B 20C 24A ··· 24H 30A 30B 60A ··· 60F order 1 2 2 2 3 3 4 4 4 4 4 4 5 6 6 6 6 6 6 8 8 8 8 10 12 12 12 12 12 12 12 ··· 12 15 15 20 20 20 24 ··· 24 30 30 60 ··· 60 size 1 1 5 5 1 1 2 4 10 20 20 20 4 1 1 5 5 5 5 10 10 10 10 4 2 2 4 4 10 10 20 ··· 20 4 4 8 8 8 10 ··· 10 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 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 SD16 Q16 C3×D4 C3×D4 C3×SD16 C3×Q16 F5 C2×F5 C3×F5 C22⋊F5 C6×F5 C3×C22⋊F5 Q8⋊F5 C3×Q8⋊F5 kernel C3×Q8⋊F5 C3×D5⋊C8 C3×C4⋊F5 C3×Q8×D5 Q8⋊F5 C3×Dic10 Q8×C15 D5⋊C8 C4⋊F5 Q8×D5 Dic10 C5×Q8 C3×Dic5 C6×D5 C3×D5 C3×D5 Dic5 D10 D5 D5 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 2 2 4 4 1 1 2 2 2 4 1 2

Matrix representation of C3×Q8⋊F5 in GL6(𝔽241)

 225 0 0 0 0 0 0 225 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
,
 1 96 0 0 0 0 5 240 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
,
 203 104 0 0 0 0 146 38 0 0 0 0 0 0 124 0 7 7 0 0 234 117 234 0 0 0 0 234 117 234 0 0 7 7 0 124
,
 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
,
 177 0 0 0 0 0 162 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))| [225,0,0,0,0,0,0,225,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,5,0,0,0,0,96,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[203,146,0,0,0,0,104,38,0,0,0,0,0,0,124,234,0,7,0,0,0,117,234,7,0,0,7,234,117,0,0,0,7,0,234,124],[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],[177,162,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;

C3×Q8⋊F5 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes F_5
% in TeX

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

G:=SmallGroup(480,289);
// by ID

G=gap.SmallGroup(480,289);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,344,2524,1271,102,9414,1595]);
// 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=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;
// generators/relations

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