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G = C158M4(2)  order 240 = 24·3·5

1st semidirect product of C15 and M4(2) acting via M4(2)/C22=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C158M4(2), Dic5.18D6, Dic5.3Dic3, C15⋊C86C2, (C2×C6).3F5, C22.(C3⋊F5), (C2×C30).2C4, C6.13(C2×F5), C30.13(C2×C4), C52(C4.Dic3), C32(C22.F5), (C2×Dic5).6S3, (C3×Dic5).5C4, C10.6(C2×Dic3), (C2×C10).2Dic3, (C6×Dic5).10C2, (C3×Dic5).24C22, C2.6(C2×C3⋊F5), SmallGroup(240,123)

Series: Derived Chief Lower central Upper central

C1C30 — C158M4(2)
C1C5C15C30C3×Dic5C15⋊C8 — C158M4(2)
C15C30 — C158M4(2)
C1C2C22

Generators and relations for C158M4(2)
 G = < a,b,c | a15=b8=c2=1, bab-1=a2, ac=ca, cbc=b5 >

2C2
5C4
5C4
2C6
2C10
5C2×C4
15C8
15C8
5C12
5C12
2C30
15M4(2)
5C3⋊C8
5C3⋊C8
5C2×C12
3C5⋊C8
3C5⋊C8
5C4.Dic3
3C22.F5

Character table of C158M4(2)

 class 12A2B34A4B4C56A6B6C8A8B8C8D10A10B10C12A12B12C12D15A15B30A30B30C30D30E30F
 size 112255104222303030304441010101044444444
ρ1111111111111111111111111111111    trivial
ρ211-1111-11-1-111-11-1-1-1111-1-111-1-11-1-11    linear of order 2
ρ311111111111-1-1-1-1111111111111111    linear of order 2
ρ411-1111-11-1-11-11-11-1-1111-1-111-1-11-1-11    linear of order 2
ρ51111-1-1-11111-i-iii111-1-1-1-111111111    linear of order 4
ρ611-11-1-111-1-11-iii-i-1-11-1-11111-1-11-1-11    linear of order 4
ρ711-11-1-111-1-11i-i-ii-1-11-1-11111-1-11-1-11    linear of order 4
ρ81111-1-1-11111ii-i-i111-1-1-1-111111111    linear of order 4
ρ922-2-122-2211-10000-2-22-1-111-1-111-111-1    orthogonal lifted from D6
ρ10222-12222-1-1-10000222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ11222-1-2-2-22-1-1-100002221111-1-1-1-1-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ1222-2-1-2-22211-10000-2-2211-1-1-1-111-111-1    symplectic lifted from Dic3, Schur index 2
ρ132-202-2i2i0200-2000000-22i-2i002200-200-2    complex lifted from M4(2)
ρ142-2022i-2i0200-2000000-2-2i2i002200-200-2    complex lifted from M4(2)
ρ152-20-1-2i2i02-3--31000000-2-ii3-3-1-1-3-31--3--31    complex lifted from C4.Dic3
ρ162-20-1-2i2i02--3-31000000-2-ii-33-1-1--3--31-3-31    complex lifted from C4.Dic3
ρ172-20-12i-2i02-3--31000000-2i-i-33-1-1-3-31--3--31    complex lifted from C4.Dic3
ρ182-20-12i-2i02--3-31000000-2i-i3-3-1-1--3--31-3-31    complex lifted from C4.Dic3
ρ1944-44000-1-4-44000011-10000-1-111-111-1    orthogonal lifted from C2×F5
ρ204444000-14440000-1-1-10000-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ214-404000-100-40000-5510000-1-1-551-551    symplectic lifted from C22.F5, Schur index 2
ρ224-404000-100-400005-510000-1-15-515-51    symplectic lifted from C22.F5, Schur index 2
ρ23444-2000-1-2-2-20000-1-1-100001+-15/21--15/21--15/21+-15/21+-15/21+-15/21--15/21--15/2    complex lifted from C3⋊F5
ρ244-40-2000-1-2-32-3200005-5100001+-15/21--15/232535232545-1--15/2353523545-1+-15/2    complex faithful
ρ254-40-2000-12-3-2-320000-55100001+-15/21--15/2354535352-1--15/232545325352-1+-15/2    complex faithful
ρ264-40-2000-1-2-32-320000-55100001--15/21+-15/232545325352-1+-15/2354535352-1--15/2    complex faithful
ρ274-40-2000-12-3-2-3200005-5100001--15/21+-15/2353523545-1+-15/232535232545-1--15/2    complex faithful
ρ28444-2000-1-2-2-20000-1-1-100001--15/21+-15/21+-15/21--15/21--15/21--15/21+-15/21+-15/2    complex lifted from C3⋊F5
ρ2944-4-2000-122-2000011-100001+-15/21--15/2-1+-15/2-1--15/21+-15/2-1--15/2-1+-15/21--15/2    complex lifted from C2×C3⋊F5
ρ3044-4-2000-122-2000011-100001--15/21+-15/2-1--15/2-1+-15/21--15/2-1+-15/2-1--15/21+-15/2    complex lifted from C2×C3⋊F5

Smallest permutation representation of C158M4(2)
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 96 40 87 28 106 53 61)(2 104 44 89 29 114 57 63)(3 97 33 76 30 107 46 65)(4 105 37 78 16 115 50 67)(5 98 41 80 17 108 54 69)(6 91 45 82 18 116 58 71)(7 99 34 84 19 109 47 73)(8 92 38 86 20 117 51 75)(9 100 42 88 21 110 55 62)(10 93 31 90 22 118 59 64)(11 101 35 77 23 111 48 66)(12 94 39 79 24 119 52 68)(13 102 43 81 25 112 56 70)(14 95 32 83 26 120 60 72)(15 103 36 85 27 113 49 74)
(61 87)(62 88)(63 89)(64 90)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(91 116)(92 117)(93 118)(94 119)(95 120)(96 106)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 113)(104 114)(105 115)

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

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

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

C158M4(2) is a maximal subgroup of
Dic5.D12  Dic5.4D12  (C2×C60).C4  C5⋊(C12.D4)  C5⋊C8.D6  S3×C22.F5  D152M4(2)  C60.59(C2×C4)  Dic10.Dic3
C158M4(2) is a maximal quotient of
C30.11C42  Dic5.13D12  C30.22M4(2)

Matrix representation of C158M4(2) in GL6(𝔽241)

24010000
24000000
005118900
0051000
00002401
000050190
,
8220000
302330000
000010
000001
0023715900
00203400
,
24000000
02400000
001000
000100
00002400
00000240

G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,240,50,0,0,0,0,1,190],[8,30,0,0,0,0,22,233,0,0,0,0,0,0,0,0,237,203,0,0,0,0,159,4,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;

C158M4(2) in GAP, Magma, Sage, TeX

C_{15}\rtimes_8M_4(2)
% in TeX

G:=Group("C15:8M4(2)");
// GroupNames label

G:=SmallGroup(240,123);
// by ID

G=gap.SmallGroup(240,123);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,50,964,5189,1745]);
// Polycyclic

G:=Group<a,b,c|a^15=b^8=c^2=1,b*a*b^-1=a^2,a*c=c*a,c*b*c=b^5>;
// generators/relations

Export

Subgroup lattice of C158M4(2) in TeX
Character table of C158M4(2) in TeX

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