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## G = C5×D4⋊2S3order 240 = 24·3·5

### Direct product of C5 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×D4⋊2S3
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C20 — C5×D4⋊2S3
 Lower central C3 — C6 — C5×D4⋊2S3
 Upper central C1 — C10 — C5×D4

Generators and relations for C5×D42S3
G = < a,b,c,d,e | a5=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 144 in 80 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, Q8, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C4○D4, C20, C20, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C5×S3, C30, C30, C2×C20, C5×D4, C5×D4, C5×Q8, D42S3, C5×Dic3, C5×Dic3, C60, S3×C10, C2×C30, C5×C4○D4, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, D4×C15, C5×D42S3
Quotients: C1, C2, C22, C5, S3, C23, C10, D6, C4○D4, C2×C10, C22×S3, C5×S3, C22×C10, D42S3, S3×C10, C5×C4○D4, S3×C2×C10, C5×D42S3

Smallest permutation representation of C5×D42S3
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 86 26 107)(2 87 27 108)(3 88 28 109)(4 89 29 110)(5 90 30 106)(6 77 116 56)(7 78 117 57)(8 79 118 58)(9 80 119 59)(10 76 120 60)(11 64 25 66)(12 65 21 67)(13 61 22 68)(14 62 23 69)(15 63 24 70)(16 74 113 53)(17 75 114 54)(18 71 115 55)(19 72 111 51)(20 73 112 52)(31 98 38 91)(32 99 39 92)(33 100 40 93)(34 96 36 94)(35 97 37 95)(41 85 48 101)(42 81 49 102)(43 82 50 103)(44 83 46 104)(45 84 47 105)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 34)(12 35)(13 31)(14 32)(15 33)(16 44)(17 45)(18 41)(19 42)(20 43)(21 37)(22 38)(23 39)(24 40)(25 36)(26 116)(27 117)(28 118)(29 119)(30 120)(46 113)(47 114)(48 115)(49 111)(50 112)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)
(1 33 45)(2 34 41)(3 35 42)(4 31 43)(5 32 44)(6 15 17)(7 11 18)(8 12 19)(9 13 20)(10 14 16)(21 111 118)(22 112 119)(23 113 120)(24 114 116)(25 115 117)(26 40 47)(27 36 48)(28 37 49)(29 38 50)(30 39 46)(51 58 67)(52 59 68)(53 60 69)(54 56 70)(55 57 66)(61 73 80)(62 74 76)(63 75 77)(64 71 78)(65 72 79)(81 88 97)(82 89 98)(83 90 99)(84 86 100)(85 87 96)(91 103 110)(92 104 106)(93 105 107)(94 101 108)(95 102 109)
(6 116)(7 117)(8 118)(9 119)(10 120)(11 115)(12 111)(13 112)(14 113)(15 114)(16 23)(17 24)(18 25)(19 21)(20 22)(31 43)(32 44)(33 45)(34 41)(35 42)(36 48)(37 49)(38 50)(39 46)(40 47)(51 65)(52 61)(53 62)(54 63)(55 64)(56 77)(57 78)(58 79)(59 80)(60 76)(66 71)(67 72)(68 73)(69 74)(70 75)(81 97)(82 98)(83 99)(84 100)(85 96)(91 103)(92 104)(93 105)(94 101)(95 102)

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

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

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

C5×D42S3 is a maximal subgroup of
D60.C22  C60.10C23  D20.24D6  C60.19C23  C15⋊2- 1+4  D30.C23  D2014D6  C5×S3×C4○D4
C5×D42S3 is a maximal quotient of
C5×D4×Dic3

75 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 6C 10A 10B 10C 10D 10E ··· 10L 10M 10N 10O 10P 12 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20L 20M ··· 20T 30A 30B 30C 30D 30E ··· 30L 60A 60B 60C 60D order 1 2 2 2 2 3 4 4 4 4 4 5 5 5 5 6 6 6 10 10 10 10 10 ··· 10 10 10 10 10 12 15 15 15 15 20 20 20 20 20 ··· 20 20 ··· 20 30 30 30 30 30 ··· 30 60 60 60 60 size 1 1 2 2 6 2 2 3 3 6 6 1 1 1 1 2 4 4 1 1 1 1 2 ··· 2 6 6 6 6 4 2 2 2 2 2 2 2 2 3 ··· 3 6 ··· 6 2 2 2 2 4 ··· 4 4 4 4 4

75 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 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D6 D6 C4○D4 C5×S3 S3×C10 S3×C10 C5×C4○D4 D4⋊2S3 C5×D4⋊2S3 kernel C5×D4⋊2S3 C5×Dic6 S3×C20 C10×Dic3 C5×C3⋊D4 D4×C15 D4⋊2S3 Dic6 C4×S3 C2×Dic3 C3⋊D4 C3×D4 C5×D4 C20 C2×C10 C15 D4 C4 C22 C3 C5 C1 # reps 1 1 1 2 2 1 4 4 4 8 8 4 1 1 2 2 4 4 8 8 1 4

Matrix representation of C5×D42S3 in GL4(𝔽61) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 50 0 0 0 50 11
,
 1 0 0 0 0 1 0 0 0 0 11 39 0 0 11 50
,
 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 1 60
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,50,50,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,11,11,0,0,39,50],[60,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,60] >;

C5×D42S3 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_2S_3
% in TeX

G:=Group("C5xD4:2S3");
// GroupNames label

G:=SmallGroup(240,170);
// by ID

G=gap.SmallGroup(240,170);
# by ID

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

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

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