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## G = C5×Q8⋊3S3order 240 = 24·3·5

### Direct product of C5 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×Q8⋊3S3
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C20 — C5×Q8⋊3S3
 Lower central C3 — C6 — C5×Q8⋊3S3
 Upper central C1 — C10 — C5×Q8

Generators and relations for C5×Q83S3
G = < a,b,c,d,e | a5=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 160 in 80 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2 [×3], C3, C4 [×3], C4, C22 [×3], C5, S3 [×3], C6, C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], Dic3, C12 [×3], D6 [×3], C15, C4○D4, C20 [×3], C20, C2×C10 [×3], C4×S3 [×3], D12 [×3], C3×Q8, C5×S3 [×3], C30, C2×C20 [×3], C5×D4 [×3], C5×Q8, Q83S3, C5×Dic3, C60 [×3], S3×C10 [×3], C5×C4○D4, S3×C20 [×3], C5×D12 [×3], Q8×C15, C5×Q83S3
Quotients: C1, C2 [×7], C22 [×7], C5, S3, C23, C10 [×7], D6 [×3], C4○D4, C2×C10 [×7], C22×S3, C5×S3, C22×C10, Q83S3, S3×C10 [×3], C5×C4○D4, S3×C2×C10, C5×Q83S3

Smallest permutation representation of C5×Q83S3
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 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)
(1 33 39)(2 34 40)(3 35 36)(4 31 37)(5 32 38)(6 13 20)(7 14 16)(8 15 17)(9 11 18)(10 12 19)(21 111 118)(22 112 119)(23 113 120)(24 114 116)(25 115 117)(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 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)
(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 37)(32 38)(33 39)(34 40)(35 36)(41 47)(42 48)(43 49)(44 50)(45 46)(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)(81 109)(82 110)(83 106)(84 107)(85 108)(91 97)(92 98)(93 99)(94 100)(95 96)

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), (1,56,26,71)(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), (1,33,39)(2,34,40)(3,35,36)(4,31,37)(5,32,38)(6,13,20)(7,14,16)(8,15,17)(9,11,18)(10,12,19)(21,111,118)(22,112,119)(23,113,120)(24,114,116)(25,115,117)(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)(81,88,109)(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), (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)(31,37)(32,38)(33,39)(34,40)(35,36)(41,47)(42,48)(43,49)(44,50)(45,46)(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)(81,109)(82,110)(83,106)(84,107)(85,108)(91,97)(92,98)(93,99)(94,100)(95,96)>;

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), (1,56,26,71)(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), (1,33,39)(2,34,40)(3,35,36)(4,31,37)(5,32,38)(6,13,20)(7,14,16)(8,15,17)(9,11,18)(10,12,19)(21,111,118)(22,112,119)(23,113,120)(24,114,116)(25,115,117)(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)(81,88,109)(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), (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)(31,37)(32,38)(33,39)(34,40)(35,36)(41,47)(42,48)(43,49)(44,50)(45,46)(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)(81,109)(82,110)(83,106)(84,107)(85,108)(91,97)(92,98)(93,99)(94,100)(95,96) );

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)], [(1,56,26,71),(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)], [(1,33,39),(2,34,40),(3,35,36),(4,31,37),(5,32,38),(6,13,20),(7,14,16),(8,15,17),(9,11,18),(10,12,19),(21,111,118),(22,112,119),(23,113,120),(24,114,116),(25,115,117),(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),(81,88,109),(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)], [(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),(31,37),(32,38),(33,39),(34,40),(35,36),(41,47),(42,48),(43,49),(44,50),(45,46),(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),(81,109),(82,110),(83,106),(84,107),(85,108),(91,97),(92,98),(93,99),(94,100),(95,96)])

C5×Q83S3 is a maximal subgroup of
D12⋊D10  Dic10.26D6  D20.27D6  Dic10.27D6  D12.29D10  D2016D6  D2017D6  C5×S3×C4○D4
C5×Q83S3 is a maximal quotient of
C5×Q8×Dic3

75 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6 10A 10B 10C 10D 10E ··· 10P 12A 12B 12C 15A 15B 15C 15D 20A ··· 20L 20M ··· 20T 30A 30B 30C 30D 60A ··· 60L order 1 2 2 2 2 3 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 6 6 6 2 2 2 2 3 3 1 1 1 1 2 1 1 1 1 6 ··· 6 4 4 4 2 2 2 2 2 ··· 2 3 ··· 3 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 D6 C4○D4 C5×S3 S3×C10 C5×C4○D4 Q8⋊3S3 C5×Q8⋊3S3 kernel C5×Q8⋊3S3 S3×C20 C5×D12 Q8×C15 Q8⋊3S3 C4×S3 D12 C3×Q8 C5×Q8 C20 C15 Q8 C4 C3 C5 C1 # reps 1 3 3 1 4 12 12 4 1 3 2 4 12 8 1 4

Matrix representation of C5×Q83S3 in GL4(𝔽61) generated by

 58 0 0 0 0 58 0 0 0 0 34 0 0 0 0 34
,
 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 50 0 0 0 0 11
,
 60 1 0 0 60 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 60
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,34,0,0,0,0,34],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,50,0,0,0,0,11],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,60] >;

C5×Q83S3 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C5xQ8:3S3");
// GroupNames label

G:=SmallGroup(240,172);
// by ID

G=gap.SmallGroup(240,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,247,794,404,194,5765]);
// Polycyclic

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

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