Copied to
clipboard

G = D30.45D4order 480 = 25·3·5

18th non-split extension by D30 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.45D4, C6.89(D4×D5), D3030(C2×C4), C10.90(S3×D4), (C2×Dic5)⋊13D6, C30.241(C2×D4), C23.D511S3, D304C432C2, C23.53(S3×D5), D153(C22⋊C4), (C2×Dic3)⋊13D10, C6.D411D5, (C22×D15)⋊10C4, (C6×Dic5)⋊6C22, (C23×D15).4C2, (C22×C10).55D6, (C22×C6).40D10, C2.7(D10⋊D6), (C2×C30).203C23, C30.151(C22×C4), C224(D30.C2), (C10×Dic3)⋊6C22, (C22×C30).65C22, (C22×D15).113C22, (C2×C6)⋊4(C4×D5), C53(S3×C22⋊C4), C32(D5×C22⋊C4), C6.56(C2×C4×D5), C10.88(S3×C2×C4), (C2×C10)⋊13(C4×S3), (C2×C30)⋊22(C2×C4), C1513(C2×C22⋊C4), C22.91(C2×S3×D5), (C2×D30.C2)⋊17C2, C2.20(C2×D30.C2), (C3×C23.D5)⋊13C2, (C5×C6.D4)⋊13C2, (C2×C6).215(C22×D5), (C2×C10).215(C22×S3), SmallGroup(480,637)

Series: Derived Chief Lower central Upper central

C1C30 — D30.45D4
C1C5C15C30C2×C30C6×Dic5C2×D30.C2 — D30.45D4
C15C30 — D30.45D4
C1C22C23

Generators and relations for D30.45D4
 G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, cac-1=dad-1=a19, cbc-1=dbd-1=a18b, dcd-1=a15c-1 >

Subgroups: 1740 in 264 conjugacy classes, 68 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C22, C22 [×2], C22 [×20], C5, S3 [×6], C6, C6 [×2], C6 [×2], C2×C4 [×8], C23, C23 [×10], D5 [×6], C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], D6 [×18], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10 [×18], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×S3 [×4], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×10], C22×C6, D15 [×4], D15 [×2], C30, C30 [×2], C30 [×2], C2×C22⋊C4, C4×D5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×D5 [×10], C22×C10, D6⋊C4 [×2], C6.D4, C3×C22⋊C4, S3×C2×C4 [×2], S3×C23, C5×Dic3 [×2], C3×Dic5 [×2], D30 [×8], D30 [×10], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C2×C4×D5 [×2], C23×D5, S3×C22⋊C4, D30.C2 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C22×D15 [×2], C22×D15 [×4], C22×D15 [×4], C22×C30, D5×C22⋊C4, D304C4 [×2], C3×C23.D5, C5×C6.D4, C2×D30.C2 [×2], C23×D15, D30.45D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D5, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4×S3 [×2], C22×S3, C2×C22⋊C4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D4 [×2], S3×D5, C2×C4×D5, D4×D5 [×2], S3×C22⋊C4, D30.C2 [×2], C2×S3×D5, D5×C22⋊C4, C2×D30.C2, D10⋊D6 [×2], D30.45D4

Smallest permutation representation of D30.45D4
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 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(55 60)(56 59)(57 58)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(113 120)(114 119)(115 118)(116 117)
(1 102 43 76)(2 91 44 65)(3 110 45 84)(4 99 46 73)(5 118 47 62)(6 107 48 81)(7 96 49 70)(8 115 50 89)(9 104 51 78)(10 93 52 67)(11 112 53 86)(12 101 54 75)(13 120 55 64)(14 109 56 83)(15 98 57 72)(16 117 58 61)(17 106 59 80)(18 95 60 69)(19 114 31 88)(20 103 32 77)(21 92 33 66)(22 111 34 85)(23 100 35 74)(24 119 36 63)(25 108 37 82)(26 97 38 71)(27 116 39 90)(28 105 40 79)(29 94 41 68)(30 113 42 87)
(1 117 16 102)(2 106 17 91)(3 95 18 110)(4 114 19 99)(5 103 20 118)(6 92 21 107)(7 111 22 96)(8 100 23 115)(9 119 24 104)(10 108 25 93)(11 97 26 112)(12 116 27 101)(13 105 28 120)(14 94 29 109)(15 113 30 98)(31 73 46 88)(32 62 47 77)(33 81 48 66)(34 70 49 85)(35 89 50 74)(36 78 51 63)(37 67 52 82)(38 86 53 71)(39 75 54 90)(40 64 55 79)(41 83 56 68)(42 72 57 87)(43 61 58 76)(44 80 59 65)(45 69 60 84)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A···20H30A···30N
order122222222222344444444556666610···101010101012121212151520···2030···30
size111122151515153030266661010101022222442···24444202020204412···124···4

66 irreducible representations

dim1111111222222222444444
type+++++++++++++++++++
imageC1C2C2C2C2C2C4S3D4D5D6D6D10D10C4×S3C4×D5S3×D4S3×D5D4×D5D30.C2C2×S3×D5D10⋊D6
kernelD30.45D4D304C4C3×C23.D5C5×C6.D4C2×D30.C2C23×D15C22×D15C23.D5D30C6.D4C2×Dic5C22×C10C2×Dic3C22×C6C2×C10C2×C6C10C23C6C22C22C2
# reps1211218142214248224428

Matrix representation of D30.45D4 in GL6(𝔽61)

43600000
100000
00605600
0025200
0000600
0000060
,
43600000
18180000
00605600
000100
0000600
0000060
,
100000
43600000
0050000
0005000
00005052
00005411
,
6000000
1810000
0011000
0001100
00005052
0000011

G:=sub<GL(6,GF(61))| [43,1,0,0,0,0,60,0,0,0,0,0,0,0,60,25,0,0,0,0,56,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[43,18,0,0,0,0,60,18,0,0,0,0,0,0,60,0,0,0,0,0,56,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,54,0,0,0,0,52,11],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,50,0,0,0,0,0,52,11] >;

D30.45D4 in GAP, Magma, Sage, TeX

D_{30}._{45}D_4
% in TeX

G:=Group("D30.45D4");
// GroupNames label

G:=SmallGroup(480,637);
// by ID

G=gap.SmallGroup(480,637);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,219,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^19,c*b*c^-1=d*b*d^-1=a^18*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

׿
×
𝔽