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G = D3016D4order 480 = 25·3·5

1st semidirect product of D30 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D3016D4, C223D60, C23.19D30, (C2×C20)⋊3D6, (C2×C4)⋊1D30, (C2×C6)⋊3D20, (C2×C30)⋊1D4, (C2×D60)⋊5C2, (C2×C12)⋊3D10, (C2×C10)⋊6D12, C2.7(D4×D15), C6.98(D4×D5), C2.7(C2×D60), C1511C22≀C2, C52(D6⋊D4), (C2×C60)⋊2C22, C22⋊C42D15, C6.33(C2×D20), D303C44C2, C32(C22⋊D20), C10.100(S3×D4), C30.306(C2×D4), C10.34(C2×D12), (C23×D15)⋊1C2, (C22×C6).57D10, (C22×C10).72D6, (C2×C30).279C23, (C2×Dic15)⋊1C22, (C22×D15)⋊1C22, (C22×C30).13C22, C22.41(C22×D15), (C2×C157D4)⋊1C2, (C5×C22⋊C4)⋊3S3, (C3×C22⋊C4)⋊3D5, (C15×C22⋊C4)⋊5C2, (C2×C6).275(C22×D5), (C2×C10).274(C22×S3), SmallGroup(480,847)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D3016D4
C1C5C15C30C2×C30C22×D15C23×D15 — D3016D4
C15C2×C30 — D3016D4
C1C22C22⋊C4

Generators and relations for D3016D4
 G = < a,b,c,d | a30=b2=c4=d2=1, bab=dad=a-1, ac=ca, cbc-1=a15b, dbd=a13b, dcd=c-1 >

Subgroups: 2084 in 260 conjugacy classes, 59 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3 [×5], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4, D4 [×6], C23, C23 [×9], D5 [×5], C10, C10 [×2], C10 [×2], Dic3, C12 [×2], D6 [×19], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, C22⋊C4 [×2], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×19], C2×C10, C2×C10 [×2], C2×C10 [×2], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C22×S3 [×9], C22×C6, D15 [×5], C30, C30 [×2], C30 [×2], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C22×D5 [×9], C22×C10, D6⋊C4 [×2], C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, S3×C23, Dic15, C60 [×2], D30 [×4], D30 [×15], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, D6⋊D4, D60 [×4], C2×Dic15, C157D4 [×2], C2×C60 [×2], C22×D15, C22×D15 [×2], C22×D15 [×6], C22×C30, C22⋊D20, D303C4 [×2], C15×C22⋊C4, C2×D60 [×2], C2×C157D4, C23×D15, D3016D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], D12 [×2], C22×S3, D15, C22≀C2, D20 [×2], C22×D5, C2×D12, S3×D4 [×2], D30 [×3], C2×D20, D4×D5 [×2], D6⋊D4, D60 [×2], C22×D15, C22⋊D20, C2×D60, D4×D15 [×2], D3016D4

Smallest permutation representation of D3016D4
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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 35)(32 34)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 90)(88 89)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
(1 106 74 56)(2 107 75 57)(3 108 76 58)(4 109 77 59)(5 110 78 60)(6 111 79 31)(7 112 80 32)(8 113 81 33)(9 114 82 34)(10 115 83 35)(11 116 84 36)(12 117 85 37)(13 118 86 38)(14 119 87 39)(15 120 88 40)(16 91 89 41)(17 92 90 42)(18 93 61 43)(19 94 62 44)(20 95 63 45)(21 96 64 46)(22 97 65 47)(23 98 66 48)(24 99 67 49)(25 100 68 50)(26 101 69 51)(27 102 70 52)(28 103 71 53)(29 104 72 54)(30 105 73 55)
(1 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 60)(28 59)(29 58)(30 57)(61 119)(62 118)(63 117)(64 116)(65 115)(66 114)(67 113)(68 112)(69 111)(70 110)(71 109)(72 108)(73 107)(74 106)(75 105)(76 104)(77 103)(78 102)(79 101)(80 100)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(90 120)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B15C15D20A···20H30A···30L30M···30T60A···60P
order122222222223444556666610···1010101010121212121515151520···2030···3030···3060···60
size11112230303030602446022222442···24444444422224···42···24···44···4

84 irreducible representations

dim11111122222222222222444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D12D15D20D30D30D60S3×D4D4×D5D4×D15
kernelD3016D4D303C4C15×C22⋊C4C2×D60C2×C157D4C23×D15C5×C22⋊C4D30C2×C30C3×C22⋊C4C2×C20C22×C10C2×C12C22×C6C2×C10C22⋊C4C2×C6C2×C4C23C22C10C6C2
# reps121211142221424488416248

Matrix representation of D3016D4 in GL6(𝔽61)

010000
60180000
00532000
0049700
0000600
0000060
,
2380000
56380000
00532000
006800
0000600
0000171
,
2540000
57360000
0060000
0006000
00004459
00002217
,
2860000
22330000
0060000
0045100
00004459
00002217

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,18,0,0,0,0,0,0,53,49,0,0,0,0,20,7,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[23,56,0,0,0,0,8,38,0,0,0,0,0,0,53,6,0,0,0,0,20,8,0,0,0,0,0,0,60,17,0,0,0,0,0,1],[25,57,0,0,0,0,4,36,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,22,0,0,0,0,59,17],[28,22,0,0,0,0,6,33,0,0,0,0,0,0,60,45,0,0,0,0,0,1,0,0,0,0,0,0,44,22,0,0,0,0,59,17] >;

D3016D4 in GAP, Magma, Sage, TeX

D_{30}\rtimes_{16}D_4
% in TeX

G:=Group("D30:16D4");
// GroupNames label

G:=SmallGroup(480,847);
// by ID

G=gap.SmallGroup(480,847);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^15*b,d*b*d=a^13*b,d*c*d=c^-1>;
// generators/relations

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