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G = D30.5C4order 240 = 24·3·5

3rd non-split extension by D30 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.5C4, C20.34D6, C157M4(2), C12.34D10, C60.34C22, Dic15.5C4, C3⋊C85D5, C52C85S3, C6.2(C4×D5), C53(C8⋊S3), C31(C8⋊D5), C10.9(C4×S3), C4.27(S3×D5), C30.27(C2×C4), (C4×D15).5C2, C2.3(D30.C2), (C5×C3⋊C8)⋊7C2, (C3×C52C8)⋊7C2, SmallGroup(240,12)

Series: Derived Chief Lower central Upper central

C1C30 — D30.5C4
C1C5C15C30C60C3×C52C8 — D30.5C4
C15C30 — D30.5C4
C1C4

Generators and relations for D30.5C4
 G = < a,b,c | a30=b2=1, c4=a15, bab=a-1, cac-1=a19, cbc-1=a3b >

30C2
15C22
15C4
10S3
6D5
3C8
5C8
15C2×C4
5Dic3
5D6
3D10
3Dic5
2D15
15M4(2)
5C24
5C4×S3
3C40
3C4×D5
5C8⋊S3
3C8⋊D5

Smallest permutation representation of D30.5C4
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 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 95)(92 94)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)
(1 116 46 76 16 101 31 61)(2 105 47 65 17 120 32 80)(3 94 48 84 18 109 33 69)(4 113 49 73 19 98 34 88)(5 102 50 62 20 117 35 77)(6 91 51 81 21 106 36 66)(7 110 52 70 22 95 37 85)(8 99 53 89 23 114 38 74)(9 118 54 78 24 103 39 63)(10 107 55 67 25 92 40 82)(11 96 56 86 26 111 41 71)(12 115 57 75 27 100 42 90)(13 104 58 64 28 119 43 79)(14 93 59 83 29 108 44 68)(15 112 60 72 30 97 45 87)

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

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

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

D30.5C4 is a maximal subgroup of
D5×C8⋊S3  S3×C8⋊D5  C40.54D6  C40.35D6  D60.5C4  D60.4C4  D154M4(2)  D30.8D4  D30.9D4  Dic6⋊D10  D125D10  D60⋊C22  C60.C23  D20.17D6  D30.44D4
D30.5C4 is a maximal quotient of
C30.23C42  D304C8  C60.14Q8

42 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B 6 8A8B8C8D10A10B12A12B15A15B20A20B20C20D24A24B24C24D30A30B40A···40H60A60B60C60D
order122344455688881010121215152020202024242424303040···4060606060
size113021130222661010222244222210101010446···64444

42 irreducible representations

dim111111222222222444
type++++++++++
imageC1C2C2C2C4C4S3D5D6M4(2)D10C4×S3C4×D5C8⋊S3C8⋊D5S3×D5D30.C2D30.5C4
kernelD30.5C4C5×C3⋊C8C3×C52C8C4×D15Dic15D30C52C8C3⋊C8C20C15C12C10C6C5C3C4C2C1
# reps111122121222448224

Matrix representation of D30.5C4 in GL4(𝔽241) generated by

1100
18818900
002401
002400
,
019000
189000
002400
002401
,
2299900
991200
002400
000240
G:=sub<GL(4,GF(241))| [1,188,0,0,1,189,0,0,0,0,240,240,0,0,1,0],[0,189,0,0,190,0,0,0,0,0,240,240,0,0,0,1],[229,99,0,0,99,12,0,0,0,0,240,0,0,0,0,240] >;

D30.5C4 in GAP, Magma, Sage, TeX

D_{30}._5C_4
% in TeX

G:=Group("D30.5C4");
// GroupNames label

G:=SmallGroup(240,12);
// by ID

G=gap.SmallGroup(240,12);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,31,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^30=b^2=1,c^4=a^15,b*a*b=a^-1,c*a*c^-1=a^19,c*b*c^-1=a^3*b>;
// generators/relations

Export

Subgroup lattice of D30.5C4 in TeX

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