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G = C245D15order 480 = 25·3·5

1st semidirect product of C24 and D15 acting via D15/C15=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C245D15, C23.31D30, (C2×C30)⋊30D4, (C23×C6)⋊5D5, C1517C22≀C2, (C23×C30)⋊3C2, (C23×C10)⋊9S3, C53(C244S3), C30.395(C2×D4), C33(C242D5), C224(C157D4), C30.38D413C2, (C2×C30).317C23, (C2×Dic15)⋊3C22, (C22×C6).123D10, (C22×C10).141D6, (C22×D15)⋊2C22, C22.66(C22×D15), (C22×C30).146C22, (C2×C157D4)⋊8C2, (C2×C6)⋊14(C5⋊D4), C6.122(C2×C5⋊D4), C2.26(C2×C157D4), (C2×C10)⋊18(C3⋊D4), C10.122(C2×C3⋊D4), (C2×C6).313(C22×D5), (C2×C10).312(C22×S3), SmallGroup(480,918)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C245D15
C1C5C15C30C2×C30C22×D15C2×C157D4 — C245D15
C15C2×C30 — C245D15
C1C22C24

Generators and relations for C245D15
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1204 in 260 conjugacy classes, 71 normal (14 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×6], C22 [×17], C5, S3, C6 [×3], C6 [×6], C2×C4 [×3], D4 [×6], C23 [×3], C23 [×7], D5, C10 [×3], C10 [×6], Dic3 [×3], D6 [×3], C2×C6, C2×C6 [×6], C2×C6 [×14], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×3], D10 [×3], C2×C10, C2×C10 [×6], C2×C10 [×14], C2×Dic3 [×3], C3⋊D4 [×6], C22×S3, C22×C6 [×3], C22×C6 [×6], D15, C30 [×3], C30 [×6], C22≀C2, C2×Dic5 [×3], C5⋊D4 [×6], C22×D5, C22×C10 [×3], C22×C10 [×6], C6.D4 [×3], C2×C3⋊D4 [×3], C23×C6, Dic15 [×3], D30 [×3], C2×C30, C2×C30 [×6], C2×C30 [×14], C23.D5 [×3], C2×C5⋊D4 [×3], C23×C10, C244S3, C2×Dic15 [×3], C157D4 [×6], C22×D15, C22×C30 [×3], C22×C30 [×6], C242D5, C30.38D4 [×3], C2×C157D4 [×3], C23×C30, C245D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×6], C22×S3, D15, C22≀C2, C5⋊D4 [×6], C22×D5, C2×C3⋊D4 [×3], D30 [×3], C2×C5⋊D4 [×3], C244S3, C157D4 [×6], C22×D15, C242D5, C2×C157D4 [×3], C245D15

Smallest permutation representation of C245D15
On 120 points
Generators in S120
(1 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(45 53)
(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 91)(73 92)(74 93)(75 94)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)
(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 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 91)(73 92)(74 93)(75 94)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 106)(85 107)(86 108)(87 109)(88 110)(89 111)(90 112)
(1 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(45 53)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 76)(70 77)(71 78)(72 79)(73 80)(74 81)(75 82)(91 116)(92 117)(93 118)(94 119)(95 120)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)
(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 65)(2 64)(3 63)(4 62)(5 61)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 84)(17 83)(18 82)(19 81)(20 80)(21 79)(22 78)(23 77)(24 76)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 105)(45 104)(46 106)(47 120)(48 119)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 110)(58 109)(59 108)(60 107)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D···2I2J 3 4A4B4C5A5B6A···6O10A···10AD15A15B15C15D30A···30BH
order12222···223444556···610···101515151530···30
size11112···2602606060222···22···222222···2

126 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D5D6D10C3⋊D4D15C5⋊D4D30C157D4
kernelC245D15C30.38D4C2×C157D4C23×C30C23×C10C2×C30C23×C6C22×C10C22×C6C2×C10C24C2×C6C23C22
# reps133116236124241248

Matrix representation of C245D15 in GL4(𝔽61) generated by

60000
0100
0010
0001
,
1000
0100
0010
00060
,
1000
0100
00600
00060
,
60000
06000
0010
0001
,
22000
02500
00200
00058
,
02500
22000
00058
00200
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[22,0,0,0,0,25,0,0,0,0,20,0,0,0,0,58],[0,22,0,0,25,0,0,0,0,0,0,20,0,0,58,0] >;

C245D15 in GAP, Magma, Sage, TeX

C_2^4\rtimes_5D_{15}
% in TeX

G:=Group("C2^4:5D15");
// GroupNames label

G:=SmallGroup(480,918);
// by ID

G=gap.SmallGroup(480,918);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,254,2693,18822]);
// Polycyclic

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

׿
×
𝔽