Copied to
clipboard

## G = C5×C24⋊C2order 240 = 24·3·5

### Direct product of C5 and C24⋊C2

Aliases: C5×C24⋊C2, C406S3, C242C10, C1208C2, C159SD16, C30.28D4, C20.52D6, Dic61C10, D12.1C10, C10.13D12, C60.64C22, C82(C5×S3), C6.1(C5×D4), C31(C5×SD16), C4.8(S3×C10), C2.3(C5×D12), C12.8(C2×C10), (C5×Dic6)⋊7C2, (C5×D12).3C2, SmallGroup(240,51)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C24⋊C2
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C5×C24⋊C2
 Lower central C3 — C6 — C12 — C5×C24⋊C2
 Upper central C1 — C10 — C20 — C40

Generators and relations for C5×C24⋊C2
G = < a,b,c | a5=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Smallest permutation representation of C5×C24⋊C2
On 120 points
Generators in S120
(1 102 26 69 73)(2 103 27 70 74)(3 104 28 71 75)(4 105 29 72 76)(5 106 30 49 77)(6 107 31 50 78)(7 108 32 51 79)(8 109 33 52 80)(9 110 34 53 81)(10 111 35 54 82)(11 112 36 55 83)(12 113 37 56 84)(13 114 38 57 85)(14 115 39 58 86)(15 116 40 59 87)(16 117 41 60 88)(17 118 42 61 89)(18 119 43 62 90)(19 120 44 63 91)(20 97 45 64 92)(21 98 46 65 93)(22 99 47 66 94)(23 100 48 67 95)(24 101 25 68 96)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 39)(27 37)(28 48)(29 35)(30 46)(31 33)(32 44)(34 42)(36 40)(41 47)(43 45)(49 65)(50 52)(51 63)(53 61)(54 72)(55 59)(56 70)(58 68)(60 66)(62 64)(67 71)(74 84)(75 95)(76 82)(77 93)(78 80)(79 91)(81 89)(83 87)(86 96)(88 94)(90 92)(97 119)(98 106)(99 117)(100 104)(101 115)(103 113)(105 111)(107 109)(108 120)(110 118)(112 116)

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

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

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

C5×C24⋊C2 is a maximal subgroup of
C24⋊D10  C4014D6  C408D6  Dic60⋊C2  C40.31D6  Dic6.D10  D30.4D4  C5×S3×SD16

75 conjugacy classes

 class 1 2A 2B 3 4A 4B 5A 5B 5C 5D 6 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40H 60A ··· 60H 120A ··· 120P order 1 2 2 3 4 4 5 5 5 5 6 8 8 10 10 10 10 10 10 10 10 12 12 15 15 15 15 20 20 20 20 20 20 20 20 24 24 24 24 30 30 30 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 12 2 2 12 1 1 1 1 2 2 2 1 1 1 1 12 12 12 12 2 2 2 2 2 2 2 2 2 2 12 12 12 12 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D4 D6 SD16 D12 C5×S3 C5×D4 C24⋊C2 S3×C10 C5×SD16 C5×D12 C5×C24⋊C2 kernel C5×C24⋊C2 C120 C5×Dic6 C5×D12 C24⋊C2 C24 Dic6 D12 C40 C30 C20 C15 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 4 4 4 4 1 1 1 2 2 4 4 4 4 8 8 16

Matrix representation of C5×C24⋊C2 in GL2(𝔽11) generated by

 5 0 0 5
,
 8 9 1 1
,
 7 4 10 4
G:=sub<GL(2,GF(11))| [5,0,0,5],[8,1,9,1],[7,10,4,4] >;

C5×C24⋊C2 in GAP, Magma, Sage, TeX

C_5\times C_{24}\rtimes C_2
% in TeX

G:=Group("C5xC24:C2");
// GroupNames label

G:=SmallGroup(240,51);
// by ID

G=gap.SmallGroup(240,51);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,265,127,1443,69,5765]);
// Polycyclic

G:=Group<a,b,c|a^5=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations

Export

׿
×
𝔽