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

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

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

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

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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