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G = C15×C22⋊C4order 240 = 24·3·5

Direct product of C15 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C15×C22⋊C4, C222C60, C30.52D4, C23.2C30, (C2×C6)⋊1C20, (C2×C60)⋊4C2, (C2×C4)⋊1C30, (C2×C20)⋊2C6, (C2×C30)⋊5C4, (C2×C12)⋊2C10, (C2×C10)⋊6C12, C2.1(C2×C60), C6.12(C5×D4), C2.1(D4×C15), C30.62(C2×C4), C6.10(C2×C20), C10.12(C3×D4), C10.17(C2×C12), (C22×C10).3C6, (C22×C30).1C2, (C22×C6).1C10, C22.2(C2×C30), (C2×C30).52C22, (C2×C6).13(C2×C10), (C2×C10).13(C2×C6), SmallGroup(240,82)

Series: Derived Chief Lower central Upper central

C1C2 — C15×C22⋊C4
C1C2C22C2×C10C2×C30C2×C60 — C15×C22⋊C4
C1C2 — C15×C22⋊C4
C1C2×C30 — C15×C22⋊C4

Generators and relations for C15×C22⋊C4
 G = < a,b,c,d | a15=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 92 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C22, C22 [×2], C22 [×2], C5, C6, C6 [×2], C6 [×2], C2×C4 [×2], C23, C10, C10 [×2], C10 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C22×C6, C30, C30 [×2], C30 [×2], C2×C20 [×2], C22×C10, C3×C22⋊C4, C60 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C5×C22⋊C4, C2×C60 [×2], C22×C30, C15×C22⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, D4 [×2], C10 [×3], C12 [×2], C2×C6, C15, C22⋊C4, C20 [×2], C2×C10, C2×C12, C3×D4 [×2], C30 [×3], C2×C20, C5×D4 [×2], C3×C22⋊C4, C60 [×2], C2×C30, C5×C22⋊C4, C2×C60, D4×C15 [×2], C15×C22⋊C4

Smallest permutation representation of C15×C22⋊C4
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)
(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 46)(30 47)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 106)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 46)(30 47)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 106)
(1 120 80 51)(2 106 81 52)(3 107 82 53)(4 108 83 54)(5 109 84 55)(6 110 85 56)(7 111 86 57)(8 112 87 58)(9 113 88 59)(10 114 89 60)(11 115 90 46)(12 116 76 47)(13 117 77 48)(14 118 78 49)(15 119 79 50)(16 94 71 32)(17 95 72 33)(18 96 73 34)(19 97 74 35)(20 98 75 36)(21 99 61 37)(22 100 62 38)(23 101 63 39)(24 102 64 40)(25 103 65 41)(26 104 66 42)(27 105 67 43)(28 91 68 44)(29 92 69 45)(30 93 70 31)

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

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

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

C15×C22⋊C4 is a maximal subgroup of
C23.6D30  C23.15D30  C222Dic30  C23.8D30  Dic1519D4  D3016D4  D30.28D4  D309D4  C23.11D30  C22.D60  D4×C60

150 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D5A5B5C5D6A···6F6G6H6I6J10A···10L10M···10T12A···12H15A···15H20A···20P30A···30X30Y···30AN60A···60AF
order12222233444455556···6666610···1010···1012···1215···1520···2030···3030···3060···60
size11112211222211111···122221···12···22···21···12···21···12···22···2

150 irreducible representations

dim11111111111111112222
type++++
imageC1C2C2C3C4C5C6C6C10C10C12C15C20C30C30C60D4C3×D4C5×D4D4×C15
kernelC15×C22⋊C4C2×C60C22×C30C5×C22⋊C4C2×C30C3×C22⋊C4C2×C20C22×C10C2×C12C22×C6C2×C10C22⋊C4C2×C6C2×C4C23C22C30C10C6C2
# reps121244428488161683224816

Matrix representation of C15×C22⋊C4 in GL4(𝔽61) generated by

13000
03400
00340
00034
,
60000
06000
00129
00060
,
1000
0100
00600
00060
,
1000
05000
002954
005932
G:=sub<GL(4,GF(61))| [13,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,29,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,50,0,0,0,0,29,59,0,0,54,32] >;

C15×C22⋊C4 in GAP, Magma, Sage, TeX

C_{15}\times C_2^2\rtimes C_4
% in TeX

G:=Group("C15xC2^2:C4");
// GroupNames label

G:=SmallGroup(240,82);
// by ID

G=gap.SmallGroup(240,82);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-2,720,745]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

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