Copied to
clipboard

## 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 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], Dic3 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C15, C2×D4, Dic5 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, C5×S3 [×2], C3×D5 [×2], C30, C30 [×2], C2×Dic5, C5⋊D4 [×4], C22×D5, C22×C10, C2×C3⋊D4, Dic15 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×C5⋊D4, C15⋊D4 [×4], C2×Dic15, D5×C2×C6, S3×C2×C10, C2×C15⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, C15⋊D4 [×2], C2×S3×D5, C2×C15⋊D4

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

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

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

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

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

׿
×
𝔽