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G = D30.C23order 480 = 25·3·5

13rd non-split extension by D30 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D608C22, Dic612D10, Dic1012D6, C30.28C24, C60.52C23, D30.13C23, Dic15.15C23, D49(S3×D5), (C4×D5)⋊9D6, C5⋊D44D6, (C4×S3)⋊9D10, (C5×D4)⋊12D6, (D4×D15)⋊5C2, C3⋊D44D10, D15⋊Q85C2, C15⋊Q83C22, (C3×D4)⋊12D10, D42D56S3, D42S36D5, D152(C4○D4), C12.28D104C2, D10⋊D63C2, D60⋊C24C2, (S3×C20)⋊3C22, (C2×Dic5)⋊15D6, (D5×C12)⋊3C22, C3⋊D205C22, C157D44C22, C5⋊D125C22, (C2×C30).4C23, C6.28(C23×D5), (C2×Dic3)⋊15D10, Dic3.D104C2, Dic5.D64C2, (D4×C15)⋊10C22, C20.52(C22×S3), C10.28(S3×C23), (C5×Dic6)⋊8C22, (C6×D5).12C23, D6.13(C22×D5), C12.52(C22×D5), (S3×C10).13C23, D30.C212C22, (C6×Dic5)⋊13C22, (D5×Dic3)⋊12C22, (C3×Dic10)⋊8C22, (S3×Dic5)⋊12C22, D10.13(C22×S3), (C4×D15).18C22, (C10×Dic3)⋊13C22, Dic3.14(C22×D5), (C5×Dic3).15C23, Dic5.14(C22×S3), (C3×Dic5).13C23, (C22×D15).74C22, (C4×S3×D5)⋊4C2, C54(S3×C4○D4), C34(D5×C4○D4), C4.52(C2×S3×D5), C1513(C2×C4○D4), C22.4(C2×S3×D5), (C3×D42D5)⋊6C2, (C5×D42S3)⋊6C2, C2.31(C22×S3×D5), (C3×C5⋊D4)⋊4C22, (C5×C3⋊D4)⋊4C22, (C2×S3×D5).10C22, (C2×C6).4(C22×D5), (C2×D30.C2)⋊21C2, (C2×C10).4(C22×S3), SmallGroup(480,1100)

Series: Derived Chief Lower central Upper central

C1C30 — D30.C23
C1C5C15C30C6×D5C2×S3×D5C4×S3×D5 — D30.C23
C15C30 — D30.C23
C1C2D4

Generators and relations for D30.C23
 G = < a,b,c,d,e | a30=b2=c2=e2=1, d2=a15, bab=eae=a-1, cac=a11, ad=da, cbc=a25b, dbd-1=a15b, ebe=a28b, cd=dc, ce=ec, de=ed >

Subgroups: 1692 in 328 conjugacy classes, 110 normal (50 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×7], C22 [×2], C22 [×11], C5, S3 [×5], C6, C6 [×3], C2×C4 [×16], D4, D4 [×11], Q8 [×4], C23 [×3], D5 [×5], C10, C10 [×3], Dic3, Dic3 [×2], Dic3, C12, C12 [×3], D6, D6 [×9], C2×C6 [×2], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5, Dic5 [×2], Dic5, C20, C20 [×3], D10, D10 [×9], C2×C10 [×2], C2×C10, Dic6, Dic6 [×2], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×2], C2×Dic3, C3⋊D4 [×2], C3⋊D4 [×4], C2×C12 [×3], C3×D4, C3×D4 [×2], C3×Q8, C22×S3 [×3], C5×S3, C3×D5, D15 [×2], D15 [×2], C30, C30 [×2], C2×C4○D4, Dic10, Dic10 [×2], C4×D5, C4×D5 [×9], D20 [×3], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×2], C5⋊D4 [×4], C2×C20 [×3], C5×D4, C5×D4 [×2], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3, D42S3 [×2], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3, C5×Dic3 [×2], C3×Dic5, C3×Dic5 [×2], Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, D30 [×2], D30 [×4], C2×C30 [×2], C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D42D5, D42D5 [×2], Q8×D5, Q82D5, C5×C4○D4, S3×C4○D4, D5×Dic3, S3×Dic5, D30.C2, D30.C2 [×6], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12, C6×Dic5 [×2], C3×C5⋊D4 [×2], C5×Dic6, S3×C20, C10×Dic3 [×2], C5×C3⋊D4 [×2], C4×D15, D60, C157D4 [×2], D4×C15, C2×S3×D5, C22×D15 [×2], D5×C4○D4, D60⋊C2, D15⋊Q8, C12.28D10, C4×S3×D5, Dic5.D6 [×2], Dic3.D10 [×2], C2×D30.C2 [×2], D10⋊D6 [×2], C3×D42D5, C5×D42S3, D4×D15, D30.C23
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D30.C23

Smallest permutation representation of D30.C23
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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(61 95)(62 94)(63 93)(64 92)(65 91)(66 120)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)(73 113)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)
(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 36)(32 47)(33 58)(34 39)(35 50)(37 42)(38 53)(40 45)(41 56)(43 48)(44 59)(46 51)(49 54)(52 57)(55 60)(61 81)(63 73)(64 84)(66 76)(67 87)(69 79)(70 90)(72 82)(75 85)(78 88)(91 96)(92 107)(93 118)(94 99)(95 110)(97 102)(98 113)(100 105)(101 116)(103 108)(104 119)(106 111)(109 114)(112 117)(115 120)
(1 86 16 71)(2 87 17 72)(3 88 18 73)(4 89 19 74)(5 90 20 75)(6 61 21 76)(7 62 22 77)(8 63 23 78)(9 64 24 79)(10 65 25 80)(11 66 26 81)(12 67 27 82)(13 68 28 83)(14 69 29 84)(15 70 30 85)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 36)(32 35)(33 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(61 66)(62 65)(63 64)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(91 96)(92 95)(93 94)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D10A10B10C10D10E10F10G10H12A12B12C12D12E15A15B20A20B20C20D20E20F20G20H20I20J30A30B30C30D30E30F60A60B
order122222222234444444444556666101010101010101012121212121515202020202020202020203030303030306060
size1122610151530302233556610103022244202244441212410102020444466661212121244888888

60 irreducible representations

dim1111111111112222222222222444448
type++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6C4○D4D10D10D10D10D10S3×D5S3×C4○D4C2×S3×D5C2×S3×D5D5×C4○D4D30.C23
kernelD30.C23D60⋊C2D15⋊Q8C12.28D10C4×S3×D5Dic5.D6Dic3.D10C2×D30.C2D10⋊D6C3×D42D5C5×D42S3D4×D15D42D5D42S3Dic10C4×D5C2×Dic5C5⋊D4C5×D4D15Dic6C4×S3C2×Dic3C3⋊D4C3×D4D4C5C4C22C3C1
# reps1111122221111211221422442222442

Matrix representation of D30.C23 in GL6(𝔽61)

6000000
0600000
00436000
001000
0000601
0000600
,
60460000
010000
00436000
00181800
0000600
0000601
,
100000
8600000
001000
000100
000001
000010
,
5000000
34110000
001000
000100
000010
000001
,
6000000
0600000
001000
00436000
000001
000010

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,43,1,0,0,0,0,60,0,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,0,0,0,0,0,46,1,0,0,0,0,0,0,43,18,0,0,0,0,60,18,0,0,0,0,0,0,60,60,0,0,0,0,0,1],[1,8,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[50,34,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D30.C23 in GAP, Magma, Sage, TeX

D_{30}.C_2^3
% in TeX

G:=Group("D30.C2^3");
// GroupNames label

G:=SmallGroup(480,1100);
// by ID

G=gap.SmallGroup(480,1100);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,346,185,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^30=b^2=c^2=e^2=1,d^2=a^15,b*a*b=e*a*e=a^-1,c*a*c=a^11,a*d=d*a,c*b*c=a^25*b,d*b*d^-1=a^15*b,e*b*e=a^28*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations

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×
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