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G = C15⋊2+ 1+4order 480 = 25·3·5

The semidirect product of C15 and 2+ 1+4 acting via 2+ 1+4/C23=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.53C24, C1582+ 1+4, D30.23C23, Dic15.25C23, C5⋊D412D6, C233(S3×D5), C15⋊Q85C22, C54(D46D6), C3⋊D415D10, (C2×Dic5)⋊9D6, (C22×D5)⋊8D6, (C22×C6)⋊7D10, D10⋊D66C2, C34(D46D10), (C2×Dic3)⋊9D10, (C22×S3)⋊7D10, (C22×C10)⋊10D6, C3⋊D207C22, C5⋊D127C22, C15⋊D47C22, C6.53(C23×D5), C30.C236C2, Dic3.D106C2, Dic5.D66C2, C157D421C22, C10.53(S3×C23), D30.C24C22, (D5×Dic3)⋊4C22, (S3×Dic5)⋊4C22, D6.21(C22×D5), (C6×D5).20C23, (S3×C10).23C23, (C2×C30).253C23, (C22×C30)⋊10C22, (C6×Dic5)⋊17C22, D10.23(C22×S3), (C2×Dic15)⋊21C22, (C10×Dic3)⋊17C22, (C22×D15)⋊16C22, Dic3.23(C22×D5), Dic5.24(C22×S3), (C5×Dic3).24C23, (C3×Dic5).22C23, (D5×C3⋊D4)⋊6C2, (S3×C5⋊D4)⋊6C2, (C2×S3×D5)⋊8C22, (C6×C5⋊D4)⋊15C2, (C2×C5⋊D4)⋊13S3, (C2×C3⋊D4)⋊13D5, (D5×C2×C6)⋊12C22, (C10×C3⋊D4)⋊15C2, (C2×C157D4)⋊26C2, C22.12(C2×S3×D5), C2.53(C22×S3×D5), (S3×C2×C10)⋊12C22, (C5×C3⋊D4)⋊16C22, (C3×C5⋊D4)⋊16C22, (C2×C6).14(C22×D5), (C2×C10).16(C22×S3), SmallGroup(480,1125)

Series: Derived Chief Lower central Upper central

C1C30 — C15⋊2+ 1+4
C1C5C15C30C6×D5C2×S3×D5D5×C3⋊D4 — C15⋊2+ 1+4
C15C30 — C15⋊2+ 1+4
C1C2C23

Generators and relations for C15⋊2+ 1+4
 G = < a,b,c,d,e | a15=b4=c2=e2=1, d2=b2, bab-1=cac=a11, dad-1=a4, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1788 in 332 conjugacy classes, 108 normal (40 characteristic)
C1, C2, C2 [×9], C3, C4 [×6], C22 [×3], C22 [×12], C5, S3 [×4], C6, C6 [×5], C2×C4 [×9], D4 [×18], Q8 [×2], C23, C23 [×5], D5 [×4], C10, C10 [×5], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6 [×3], C2×C6 [×4], C15, C2×D4 [×9], C4○D4 [×6], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10 [×3], C2×C10 [×4], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3, C2×Dic3 [×3], C3⋊D4 [×4], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×S3 [×3], C22×C6, C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C30 [×3], 2+ 1+4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5, C2×Dic5 [×3], C5⋊D4 [×4], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×D5 [×3], C22×C10, C22×C10, C4○D12 [×2], S3×D4 [×4], D42S3 [×4], C2×C3⋊D4, C2×C3⋊D4 [×3], C6×D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], S3×D5 [×2], C6×D5 [×2], C6×D5, S3×C10 [×2], S3×C10, D30 [×2], D30, C2×C30 [×3], C2×C30, C4○D20 [×2], D4×D5 [×4], D42D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×3], D4×C10, D46D6, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C6×Dic5, C3×C5⋊D4 [×4], C10×Dic3, C5×C3⋊D4 [×4], C2×Dic15, C157D4 [×4], C2×S3×D5 [×2], D5×C2×C6, S3×C2×C10, C22×D15, C22×C30, D46D10, Dic5.D6 [×2], C30.C23 [×2], Dic3.D10 [×2], D5×C3⋊D4 [×2], S3×C5⋊D4 [×2], D10⋊D6 [×2], C6×C5⋊D4, C10×C3⋊D4, C2×C157D4, C15⋊2+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, D46D6, C2×S3×D5 [×3], D46D10, C22×S3×D5, C15⋊2+ 1+4

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

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

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

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

63 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D30A···30N
order12222222222344444455666666610···101010101010101010121215152020202030···30
size11222661010303026610103030222224420202···2444412121212202044121212124···4

63 irreducible representations

dim11111111112222222222444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D10D10D10D102+ 1+4S3×D5D46D6C2×S3×D5D46D10C15⋊2+ 1+4
kernelC15⋊2+ 1+4Dic5.D6C30.C23Dic3.D10D5×C3⋊D4S3×C5⋊D4D10⋊D6C6×C5⋊D4C10×C3⋊D4C2×C157D4C2×C5⋊D4C2×C3⋊D4C2×Dic5C5⋊D4C22×D5C22×C10C2×Dic3C3⋊D4C22×S3C22×C6C15C23C5C22C3C1
# reps12222221111214112822122648

Matrix representation of C15⋊2+ 1+4 in GL4(𝔽61) generated by

8600
8000
53564838
44352323
,
00601
11465917
450150
350150
,
00601
11465917
1842150
1942150
,
81400
525300
401445
36343947
,
304500
603100
401445
11571647
G:=sub<GL(4,GF(61))| [8,8,53,44,6,0,56,35,0,0,48,23,0,0,38,23],[0,11,4,3,0,46,50,50,60,59,15,15,1,17,0,0],[0,11,18,19,0,46,42,42,60,59,15,15,1,17,0,0],[8,52,4,36,14,53,0,34,0,0,14,39,0,0,45,47],[30,60,4,11,45,31,0,57,0,0,14,16,0,0,45,47] >;

C15⋊2+ 1+4 in GAP, Magma, Sage, TeX

C_{15}\rtimes 2_+^{1+4}
% in TeX

G:=Group("C15:ES+(2,2)");
// GroupNames label

G:=SmallGroup(480,1125);
// by ID

G=gap.SmallGroup(480,1125);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^15=b^4=c^2=e^2=1,d^2=b^2,b*a*b^-1=c*a*c=a^11,d*a*d^-1=a^4,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations

׿
×
𝔽