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## G = C2×C15⋊7D4order 240 = 24·3·5

### Direct product of C2 and C15⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C15⋊7D4
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C2×C15⋊7D4
 Lower central C15 — C30 — C2×C15⋊7D4
 Upper central C1 — C22 — C23

Generators and relations for C2×C157D4
G = < a,b,c,d | a2=b15=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 520 in 108 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C6 [×2], C2×C4, D4 [×4], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], Dic3 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C2×D4, Dic5 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, D15 [×2], C30, C30 [×2], C30 [×2], C2×Dic5, C5⋊D4 [×4], C22×D5, C22×C10, C2×C3⋊D4, Dic15 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C5⋊D4, C2×Dic15, C157D4 [×4], C22×D15, C22×C30, C2×C157D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, D30 [×3], C2×C5⋊D4, C157D4 [×2], C22×D15, C2×C157D4

Smallest permutation representation of C2×C157D4
On 120 points
Generators in S120
(1 72)(2 73)(3 74)(4 75)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 101)(32 102)(33 103)(34 104)(35 105)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 106)(58 107)(59 108)(60 109)
(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 41 30 49)(2 40 16 48)(3 39 17 47)(4 38 18 46)(5 37 19 60)(6 36 20 59)(7 35 21 58)(8 34 22 57)(9 33 23 56)(10 32 24 55)(11 31 25 54)(12 45 26 53)(13 44 27 52)(14 43 28 51)(15 42 29 50)(61 92 87 109)(62 91 88 108)(63 105 89 107)(64 104 90 106)(65 103 76 120)(66 102 77 119)(67 101 78 118)(68 100 79 117)(69 99 80 116)(70 98 81 115)(71 97 82 114)(72 96 83 113)(73 95 84 112)(74 94 85 111)(75 93 86 110)
(1 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 75)(14 74)(15 73)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 84)(30 83)(31 108)(32 107)(33 106)(34 120)(35 119)(36 118)(37 117)(38 116)(39 115)(40 114)(41 113)(42 112)(43 111)(44 110)(45 109)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 105)(56 104)(57 103)(58 102)(59 101)(60 100)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D5 D6 D10 C3⋊D4 D15 C5⋊D4 D30 C15⋊7D4 kernel C2×C15⋊7D4 C2×Dic15 C15⋊7D4 C22×D15 C22×C30 C22×C10 C30 C22×C6 C2×C10 C2×C6 C10 C23 C6 C22 C2 # reps 1 1 4 1 1 1 2 2 3 6 4 4 8 12 16

Matrix representation of C2×C157D4 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 18 18 0 0 43 60 0 0 0 0 47 31 0 0 30 25
,
 1 0 0 0 43 60 0 0 0 0 37 8 0 0 27 24
,
 60 0 0 0 18 1 0 0 0 0 25 33 0 0 31 36
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[18,43,0,0,18,60,0,0,0,0,47,30,0,0,31,25],[1,43,0,0,0,60,0,0,0,0,37,27,0,0,8,24],[60,18,0,0,0,1,0,0,0,0,25,31,0,0,33,36] >;

C2×C157D4 in GAP, Magma, Sage, TeX

C_2\times C_{15}\rtimes_7D_4
% in TeX

G:=Group("C2xC15:7D4");
// GroupNames label

G:=SmallGroup(240,184);
// by ID

G=gap.SmallGroup(240,184);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,964,6917]);
// 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=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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