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## G = C5×C32⋊2Q8order 360 = 23·32·5

### Direct product of C5 and C32⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C5×C32⋊2Q8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — Dic3×C15 — C5×C32⋊2Q8
 Lower central C32 — C3×C6 — C5×C32⋊2Q8
 Upper central C1 — C10

Generators and relations for C5×C322Q8
G = < a,b,c,d,e | a5=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 132 in 54 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C4, C5, C6, C6, Q8, C32, C10, Dic3, Dic3, C12, C15, C15, C3×C6, C20, Dic6, C30, C30, C3×Dic3, C3⋊Dic3, C5×Q8, C3×C15, C5×Dic3, C5×Dic3, C60, C322Q8, C3×C30, C5×Dic6, Dic3×C15, C5×C3⋊Dic3, C5×C322Q8
Quotients: C1, C2, C22, C5, S3, Q8, C10, D6, C2×C10, Dic6, C5×S3, S32, C5×Q8, S3×C10, C322Q8, C5×Dic6, C5×S32, C5×C322Q8

Smallest permutation representation of C5×C322Q8
On 120 points
Generators in S120
(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 33 39)(2 34 40)(3 35 36)(4 31 37)(5 32 38)(6 20 13)(7 16 14)(8 17 15)(9 18 11)(10 19 12)(21 118 111)(22 119 112)(23 120 113)(24 116 114)(25 117 115)(26 47 41)(27 48 42)(28 49 43)(29 50 44)(30 46 45)(51 79 58)(52 80 59)(53 76 60)(54 77 56)(55 78 57)(61 74 67)(62 75 68)(63 71 69)(64 72 70)(65 73 66)(81 88 109)(82 89 110)(83 90 106)(84 86 107)(85 87 108)(91 97 104)(92 98 105)(93 99 101)(94 100 102)(95 96 103)
(1 33 39)(2 34 40)(3 35 36)(4 31 37)(5 32 38)(6 20 13)(7 16 14)(8 17 15)(9 18 11)(10 19 12)(21 118 111)(22 119 112)(23 120 113)(24 116 114)(25 117 115)(26 47 41)(27 48 42)(28 49 43)(29 50 44)(30 46 45)(51 58 79)(52 59 80)(53 60 76)(54 56 77)(55 57 78)(61 67 74)(62 68 75)(63 69 71)(64 70 72)(65 66 73)(81 109 88)(82 110 89)(83 106 90)(84 107 86)(85 108 87)(91 104 97)(92 105 98)(93 101 99)(94 102 100)(95 103 96)
(1 56 26 71)(2 57 27 72)(3 58 28 73)(4 59 29 74)(5 60 30 75)(6 109 21 95)(7 110 22 91)(8 106 23 92)(9 107 24 93)(10 108 25 94)(11 84 114 99)(12 85 115 100)(13 81 111 96)(14 82 112 97)(15 83 113 98)(16 89 119 104)(17 90 120 105)(18 86 116 101)(19 87 117 102)(20 88 118 103)(31 80 50 61)(32 76 46 62)(33 77 47 63)(34 78 48 64)(35 79 49 65)(36 51 43 66)(37 52 44 67)(38 53 45 68)(39 54 41 69)(40 55 42 70)
(1 86 26 101)(2 87 27 102)(3 88 28 103)(4 89 29 104)(5 90 30 105)(6 65 21 79)(7 61 22 80)(8 62 23 76)(9 63 24 77)(10 64 25 78)(11 69 114 54)(12 70 115 55)(13 66 111 51)(14 67 112 52)(15 68 113 53)(16 74 119 59)(17 75 120 60)(18 71 116 56)(19 72 117 57)(20 73 118 58)(31 110 50 91)(32 106 46 92)(33 107 47 93)(34 108 48 94)(35 109 49 95)(36 81 43 96)(37 82 44 97)(38 83 45 98)(39 84 41 99)(40 85 42 100)

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

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

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

75 conjugacy classes

 class 1 2 3A 3B 3C 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 10A 10B 10C 10D 12A 12B 12C 12D 15A ··· 15H 15I 15J 15K 15L 20A ··· 20H 20I 20J 20K 20L 30A ··· 30H 30I 30J 30K 30L 60A ··· 60P order 1 2 3 3 3 4 4 4 5 5 5 5 6 6 6 10 10 10 10 12 12 12 12 15 ··· 15 15 15 15 15 20 ··· 20 20 20 20 20 30 ··· 30 30 30 30 30 60 ··· 60 size 1 1 2 2 4 6 6 18 1 1 1 1 2 2 4 1 1 1 1 6 6 6 6 2 ··· 2 4 4 4 4 6 ··· 6 18 18 18 18 2 ··· 2 4 4 4 4 6 ··· 6

75 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + - + - image C1 C2 C2 C5 C10 C10 S3 Q8 D6 Dic6 C5×S3 C5×Q8 S3×C10 C5×Dic6 S32 C32⋊2Q8 C5×S32 C5×C32⋊2Q8 kernel C5×C32⋊2Q8 Dic3×C15 C5×C3⋊Dic3 C32⋊2Q8 C3×Dic3 C3⋊Dic3 C5×Dic3 C3×C15 C30 C15 Dic3 C32 C6 C3 C10 C5 C2 C1 # reps 1 2 1 4 8 4 2 1 2 4 8 4 8 16 1 1 4 4

Matrix representation of C5×C322Q8 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 58 0 0 0 0 0 0 58 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 53 33 0 0 0 0 35 8 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 60 60
,
 27 3 0 0 0 0 21 34 0 0 0 0 0 0 1 0 0 0 0 0 60 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[53,35,0,0,0,0,33,8,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60],[27,21,0,0,0,0,3,34,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C5×C322Q8 in GAP, Magma, Sage, TeX

C_5\times C_3^2\rtimes_2Q_8
% in TeX

G:=Group("C5xC3^2:2Q8");
// GroupNames label

G:=SmallGroup(360,76);
// by ID

G=gap.SmallGroup(360,76);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,120,265,127,1210,8645]);
// Polycyclic

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

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