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## G = C5×S3×Q8order 240 = 24·3·5

### Direct product of C5, S3 and Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×S3×Q8
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C20 — C5×S3×Q8
 Lower central C3 — C6 — C5×S3×Q8
 Upper central C1 — C10 — C5×Q8

Generators and relations for C5×S3×Q8
G = < a,b,c,d,e | a5=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 128 in 76 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2 [×2], C3, C4 [×3], C4 [×3], C22, C5, S3 [×2], C6, C2×C4 [×3], Q8, Q8 [×3], C10, C10 [×2], Dic3 [×3], C12 [×3], D6, C15, C2×Q8, C20 [×3], C20 [×3], C2×C10, Dic6 [×3], C4×S3 [×3], C3×Q8, C5×S3 [×2], C30, C2×C20 [×3], C5×Q8, C5×Q8 [×3], S3×Q8, C5×Dic3 [×3], C60 [×3], S3×C10, Q8×C10, C5×Dic6 [×3], S3×C20 [×3], Q8×C15, C5×S3×Q8
Quotients: C1, C2 [×7], C22 [×7], C5, S3, Q8 [×2], C23, C10 [×7], D6 [×3], C2×Q8, C2×C10 [×7], C22×S3, C5×S3, C5×Q8 [×2], C22×C10, S3×Q8, S3×C10 [×3], Q8×C10, S3×C2×C10, C5×S3×Q8

Smallest permutation representation of C5×S3×Q8
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 39 33)(2 40 34)(3 36 35)(4 37 31)(5 38 32)(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 41 47)(27 42 48)(28 43 49)(29 44 50)(30 45 46)(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)(81 109 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 26)(2 27)(3 28)(4 29)(5 30)(6 111)(7 112)(8 113)(9 114)(10 115)(11 24)(12 25)(13 21)(14 22)(15 23)(16 119)(17 120)(18 116)(19 117)(20 118)(31 44)(32 45)(33 41)(34 42)(35 43)(36 49)(37 50)(38 46)(39 47)(40 48)(51 65)(52 61)(53 62)(54 63)(55 64)(56 71)(57 72)(58 73)(59 74)(60 75)(66 79)(67 80)(68 76)(69 77)(70 78)(81 95)(82 91)(83 92)(84 93)(85 94)(86 101)(87 102)(88 103)(89 104)(90 105)(96 109)(97 110)(98 106)(99 107)(100 108)
(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), (1,39,33)(2,40,34)(3,36,35)(4,37,31)(5,38,32)(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)(26,41,47)(27,42,48)(28,43,49)(29,44,50)(30,45,46)(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)(81,109,88)(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), (1,26)(2,27)(3,28)(4,29)(5,30)(6,111)(7,112)(8,113)(9,114)(10,115)(11,24)(12,25)(13,21)(14,22)(15,23)(16,119)(17,120)(18,116)(19,117)(20,118)(31,44)(32,45)(33,41)(34,42)(35,43)(36,49)(37,50)(38,46)(39,47)(40,48)(51,65)(52,61)(53,62)(54,63)(55,64)(56,71)(57,72)(58,73)(59,74)(60,75)(66,79)(67,80)(68,76)(69,77)(70,78)(81,95)(82,91)(83,92)(84,93)(85,94)(86,101)(87,102)(88,103)(89,104)(90,105)(96,109)(97,110)(98,106)(99,107)(100,108), (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), (1,39,33)(2,40,34)(3,36,35)(4,37,31)(5,38,32)(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)(26,41,47)(27,42,48)(28,43,49)(29,44,50)(30,45,46)(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)(81,109,88)(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), (1,26)(2,27)(3,28)(4,29)(5,30)(6,111)(7,112)(8,113)(9,114)(10,115)(11,24)(12,25)(13,21)(14,22)(15,23)(16,119)(17,120)(18,116)(19,117)(20,118)(31,44)(32,45)(33,41)(34,42)(35,43)(36,49)(37,50)(38,46)(39,47)(40,48)(51,65)(52,61)(53,62)(54,63)(55,64)(56,71)(57,72)(58,73)(59,74)(60,75)(66,79)(67,80)(68,76)(69,77)(70,78)(81,95)(82,91)(83,92)(84,93)(85,94)(86,101)(87,102)(88,103)(89,104)(90,105)(96,109)(97,110)(98,106)(99,107)(100,108), (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)], [(1,39,33),(2,40,34),(3,36,35),(4,37,31),(5,38,32),(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),(26,41,47),(27,42,48),(28,43,49),(29,44,50),(30,45,46),(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),(81,109,88),(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)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,111),(7,112),(8,113),(9,114),(10,115),(11,24),(12,25),(13,21),(14,22),(15,23),(16,119),(17,120),(18,116),(19,117),(20,118),(31,44),(32,45),(33,41),(34,42),(35,43),(36,49),(37,50),(38,46),(39,47),(40,48),(51,65),(52,61),(53,62),(54,63),(55,64),(56,71),(57,72),(58,73),(59,74),(60,75),(66,79),(67,80),(68,76),(69,77),(70,78),(81,95),(82,91),(83,92),(84,93),(85,94),(86,101),(87,102),(88,103),(89,104),(90,105),(96,109),(97,110),(98,106),(99,107),(100,108)], [(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)])

C5×S3×Q8 is a maximal subgroup of   D20.28D6  C60.44C23  D20.29D6  C30.33C24

75 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6 10A 10B 10C 10D 10E ··· 10L 12A 12B 12C 15A 15B 15C 15D 20A ··· 20L 20M ··· 20X 30A 30B 30C 30D 60A ··· 60L order 1 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 10 10 10 10 10 ··· 10 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 30 30 30 60 ··· 60 size 1 1 3 3 2 2 2 2 6 6 6 1 1 1 1 2 1 1 1 1 3 ··· 3 4 4 4 2 2 2 2 2 ··· 2 6 ··· 6 2 2 2 2 4 ··· 4

75 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 C5 C10 C10 C10 S3 Q8 D6 C5×S3 C5×Q8 S3×C10 S3×Q8 C5×S3×Q8 kernel C5×S3×Q8 C5×Dic6 S3×C20 Q8×C15 S3×Q8 Dic6 C4×S3 C3×Q8 C5×Q8 C5×S3 C20 Q8 S3 C4 C5 C1 # reps 1 3 3 1 4 12 12 4 1 2 3 4 8 12 1 4

Matrix representation of C5×S3×Q8 in GL4(𝔽61) generated by

 9 0 0 0 0 9 0 0 0 0 58 0 0 0 0 58
,
 0 60 0 0 1 60 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 0 1 0 0 60 0
,
 60 0 0 0 0 60 0 0 0 0 13 47 0 0 47 48
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,58,0,0,0,0,58],[0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,13,47,0,0,47,48] >;

C5×S3×Q8 in GAP, Magma, Sage, TeX

C_5\times S_3\times Q_8
% in TeX

G:=Group("C5xS3xQ8");
// GroupNames label

G:=SmallGroup(240,171);
// by ID

G=gap.SmallGroup(240,171);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,404,194,5765]);
// Polycyclic

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

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