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## G = C2×C15⋊D4order 240 = 24·3·5

### Direct product of C2 and C15⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C15⋊D4
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C15⋊D4 — C2×C15⋊D4
 Lower central C15 — C30 — C2×C15⋊D4
 Upper central C1 — C22

Generators and relations for C2×C15⋊D4
G = < a,b,c,d | a2=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b4, dcd=c-1 >

Subgroups: 432 in 108 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, D5, C10, C10, C10, Dic3, D6, D6, C2×C6, C2×C6, C15, C2×D4, Dic5, D10, D10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C5×S3, C3×D5, C30, C30, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C2×C3⋊D4, Dic15, C6×D5, C6×D5, S3×C10, S3×C10, C2×C30, C2×C5⋊D4, C15⋊D4, C2×Dic15, D5×C2×C6, S3×C2×C10, C2×C15⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C5⋊D4, C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, C15⋊D4, C2×S3×D5, C2×C15⋊D4

Smallest permutation representation of C2×C15⋊D4
On 120 points
Generators in S120
(1 41)(2 42)(3 43)(4 44)(5 45)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 60)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(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)(76 115)(77 116)(78 117)(79 118)(80 119)(81 120)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 113)(90 114)
(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 61 16 89)(2 75 17 88)(3 74 18 87)(4 73 19 86)(5 72 20 85)(6 71 21 84)(7 70 22 83)(8 69 23 82)(9 68 24 81)(10 67 25 80)(11 66 26 79)(12 65 27 78)(13 64 28 77)(14 63 29 76)(15 62 30 90)(31 101 50 108)(32 100 51 107)(33 99 52 106)(34 98 53 120)(35 97 54 119)(36 96 55 118)(37 95 56 117)(38 94 57 116)(39 93 58 115)(40 92 59 114)(41 91 60 113)(42 105 46 112)(43 104 47 111)(44 103 48 110)(45 102 49 109)
(1 41)(2 45)(3 34)(4 38)(5 42)(6 31)(7 35)(8 39)(9 43)(10 32)(11 36)(12 40)(13 44)(14 33)(15 37)(16 60)(17 49)(18 53)(19 57)(20 46)(21 50)(22 54)(23 58)(24 47)(25 51)(26 55)(27 59)(28 48)(29 52)(30 56)(61 113)(62 117)(63 106)(64 110)(65 114)(66 118)(67 107)(68 111)(69 115)(70 119)(71 108)(72 112)(73 116)(74 120)(75 109)(76 99)(77 103)(78 92)(79 96)(80 100)(81 104)(82 93)(83 97)(84 101)(85 105)(86 94)(87 98)(88 102)(89 91)(90 95)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 15A 15B 30A ··· 30F order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 15 15 30 ··· 30 size 1 1 1 1 6 6 10 10 2 30 30 2 2 2 2 2 10 10 10 10 2 ··· 2 6 ··· 6 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 S3 D4 D5 D6 D6 D10 D10 C3⋊D4 C5⋊D4 S3×D5 C15⋊D4 C2×S3×D5 kernel C2×C15⋊D4 C15⋊D4 C2×Dic15 D5×C2×C6 S3×C2×C10 C22×D5 C30 C22×S3 D10 C2×C10 D6 C2×C6 C10 C6 C22 C2 C2 # reps 1 4 1 1 1 1 2 2 2 1 4 2 4 8 2 4 2

Matrix representation of C2×C15⋊D4 in GL5(𝔽61)

 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 47 0 0 0 0 27 13 0 0 0 0 0 44 44 0 0 0 17 60
,
 1 0 0 0 0 0 15 15 0 0 0 50 46 0 0 0 0 0 1 0 0 0 0 17 60
,
 1 0 0 0 0 0 60 0 0 0 0 2 1 0 0 0 0 0 1 0 0 0 0 17 60

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,47,27,0,0,0,0,13,0,0,0,0,0,44,17,0,0,0,44,60],[1,0,0,0,0,0,15,50,0,0,0,15,46,0,0,0,0,0,1,17,0,0,0,0,60],[1,0,0,0,0,0,60,2,0,0,0,0,1,0,0,0,0,0,1,17,0,0,0,0,60] >;

C2×C15⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes D_4
% in TeX

G:=Group("C2xC15:D4");
// GroupNames label

G:=SmallGroup(240,145);
// by ID

G=gap.SmallGroup(240,145);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,490,6917]);
// Polycyclic

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

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