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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C15×C22⋊C4
 Chief series C1 — C2 — C22 — C2×C10 — C2×C30 — C2×C60 — C15×C22⋊C4
 Lower central C1 — C2 — C15×C22⋊C4
 Upper central C1 — C2×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 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 5C 5D 6A ··· 6F 6G 6H 6I 6J 10A ··· 10L 10M ··· 10T 12A ··· 12H 15A ··· 15H 20A ··· 20P 30A ··· 30X 30Y ··· 30AN 60A ··· 60AF order 1 2 2 2 2 2 3 3 4 4 4 4 5 5 5 5 6 ··· 6 6 6 6 6 10 ··· 10 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 1 1 2 2 2 2 1 1 1 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C4 C5 C6 C6 C10 C10 C12 C15 C20 C30 C30 C60 D4 C3×D4 C5×D4 D4×C15 kernel C15×C22⋊C4 C2×C60 C22×C30 C5×C22⋊C4 C2×C30 C3×C22⋊C4 C2×C20 C22×C10 C2×C12 C22×C6 C2×C10 C22⋊C4 C2×C6 C2×C4 C23 C22 C30 C10 C6 C2 # reps 1 2 1 2 4 4 4 2 8 4 8 8 16 16 8 32 2 4 8 16

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

 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 29 0 0 0 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 54 0 0 59 32
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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