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## G = Q8×C3⋊F5order 480 = 25·3·5

### Direct product of Q8 and C3⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Q8×C3⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C2×C3⋊F5 — C4×C3⋊F5 — Q8×C3⋊F5
 Lower central C15 — C30 — Q8×C3⋊F5
 Upper central C1 — C2 — Q8

Generators and relations for Q8×C3⋊F5
G = < a,b,c,d,e | a4=c3=d5=e4=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 636 in 140 conjugacy classes, 63 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, Q8, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, F5, D10, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C3×D5, C30, C4×Q8, Dic10, C4×D5, C5×Q8, C2×F5, C4×Dic3, C4⋊Dic3, C6×Q8, C3×Dic5, C60, C3⋊F5, C3⋊F5, C6×D5, C4×F5, C4⋊F5, Q8×D5, Q8×Dic3, C3×Dic10, D5×C12, Q8×C15, C2×C3⋊F5, C2×C3⋊F5, Q8×F5, C4×C3⋊F5, C60⋊C4, C3×Q8×D5, Q8×C3⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, Dic3, D6, C22×C4, C2×Q8, C4○D4, F5, C2×Dic3, C22×S3, C4×Q8, C2×F5, S3×Q8, Q83S3, C22×Dic3, C3⋊F5, C22×F5, Q8×Dic3, C2×C3⋊F5, Q8×F5, C22×C3⋊F5, Q8×C3⋊F5

Smallest permutation representation of Q8×C3⋊F5
On 120 points
Generators in S120
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 91 76 106)(62 92 77 107)(63 93 78 108)(64 94 79 109)(65 95 80 110)(66 96 81 111)(67 97 82 112)(68 98 83 113)(69 99 84 114)(70 100 85 115)(71 101 86 116)(72 102 87 117)(73 103 88 118)(74 104 89 119)(75 105 90 120)
(1 79 19 64)(2 80 20 65)(3 76 16 61)(4 77 17 62)(5 78 18 63)(6 81 21 66)(7 82 22 67)(8 83 23 68)(9 84 24 69)(10 85 25 70)(11 86 26 71)(12 87 27 72)(13 88 28 73)(14 89 29 74)(15 90 30 75)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(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 19)(2 16 5 17)(3 18 4 20)(6 28 7 30)(8 27 10 26)(9 29)(11 23 12 25)(13 22 15 21)(14 24)(31 48 32 50)(33 47 35 46)(34 49)(36 58 37 60)(38 57 40 56)(39 59)(41 53 42 55)(43 52 45 51)(44 54)(61 78 62 80)(63 77 65 76)(64 79)(66 88 67 90)(68 87 70 86)(69 89)(71 83 72 85)(73 82 75 81)(74 84)(91 108 92 110)(93 107 95 106)(94 109)(96 118 97 120)(98 117 100 116)(99 119)(101 113 102 115)(103 112 105 111)(104 114)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 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 C4 C4 S3 Q8 Dic3 D6 Dic3 C4○D4 F5 C2×F5 S3×Q8 Q8⋊3S3 C3⋊F5 C2×C3⋊F5 Q8×F5 Q8×C3⋊F5 kernel Q8×C3⋊F5 C4×C3⋊F5 C60⋊C4 C3×Q8×D5 C3×Dic10 Q8×C15 Q8×D5 C3⋊F5 Dic10 C4×D5 C5×Q8 C3×D5 C3×Q8 C12 D5 D5 Q8 C4 C3 C1 # reps 1 3 3 1 6 2 1 2 3 3 1 2 1 3 1 1 2 6 1 2

Matrix representation of Q8×C3⋊F5 in GL6(𝔽61)

 60 21 0 0 0 0 58 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 40 33 0 0 0 0 55 21 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 0 0 0 0 0 0 1 0 0 0 0 0 0 33 55 0 6 0 0 0 27 55 6 0 0 6 55 27 0 0 0 6 0 55 33
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 60 0 1 0 0 0 60 0 0 1 0 0 60 0 0 0
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 1 0 60 0 0 0 0 0 60 1 0 0 0 1 60 0 0 0 0 0 60 0

G:=sub<GL(6,GF(61))| [60,58,0,0,0,0,21,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[40,55,0,0,0,0,33,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,0,6,6,0,0,55,27,55,0,0,0,0,55,27,55,0,0,6,6,0,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,60,60,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60,0,0,0,1,0,0] >;

Q8×C3⋊F5 in GAP, Magma, Sage, TeX

Q_8\times C_3\rtimes F_5
% in TeX

G:=Group("Q8xC3:F5");
// GroupNames label

G:=SmallGroup(480,1069);
// by ID

G=gap.SmallGroup(480,1069);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,219,100,2693,14118,2379]);
// Polycyclic

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

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