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G = D154M4(2)  order 480 = 25·3·5

The semidirect product of D15 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D154M4(2), C60.185C23, C3⋊C822D10, C52C822D6, C57(S3×M4(2)), C33(D5×M4(2)), C20.54(C4×S3), (C2×C20).82D6, C12.22(C4×D5), C4.Dic56S3, C4.Dic36D5, C60.131(C2×C4), D30.37(C2×C4), (C4×D15).10C4, (C2×C12).83D10, C1517(C2×M4(2)), D152C813C2, D30.5C414C2, C30.107(C22×C4), (C2×C60).215C22, C20.182(C22×S3), C4.14(D30.C2), (C2×Dic15).22C4, Dic15.45(C2×C4), (C22×D15).13C4, (C4×D15).65C22, C12.182(C22×D5), C22.6(D30.C2), C6.41(C2×C4×D5), C10.74(S3×C2×C4), C4.155(C2×S3×D5), (C5×C3⋊C8)⋊22C22, (C2×C4×D15).18C2, (C2×C6).10(C4×D5), (C2×C10).33(C4×S3), (C2×C4).195(S3×D5), C2.6(C2×D30.C2), (C2×C30).104(C2×C4), (C3×C52C8)⋊22C22, (C5×C4.Dic3)⋊12C2, (C3×C4.Dic5)⋊12C2, SmallGroup(480,368)

Series: Derived Chief Lower central Upper central

C1C30 — D154M4(2)
C1C5C15C30C60C3×C52C8D152C8 — D154M4(2)
C15C30 — D154M4(2)
C1C4C2×C4

Generators and relations for D154M4(2)
 G = < a,b,c,d | a15=b2=c8=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, bd=db, dcd=c5 >

Subgroups: 668 in 136 conjugacy classes, 54 normal (40 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×3], C6, C6, C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×3], C10, C10, Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, D15 [×2], D15, C30, C30, C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, Dic15 [×2], C60 [×2], D30 [×2], D30 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), C2×C4×D5, S3×M4(2), C5×C3⋊C8 [×2], C3×C52C8 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C22×D15, D5×M4(2), D152C8 [×2], D30.5C4 [×2], C3×C4.Dic5, C5×C4.Dic3, C2×C4×D15, D154M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C2×M4(2), C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, S3×M4(2), D30.C2 [×2], C2×S3×D5, D5×M4(2), C2×D30.C2, D154M4(2)

Smallest permutation representation of D154M4(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(76 77)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(91 101)(92 100)(93 99)(94 98)(95 97)(102 105)(103 104)(106 108)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
(1 115 43 68 28 104 46 77)(2 111 44 64 29 100 47 88)(3 107 45 75 30 96 48 84)(4 118 31 71 16 92 49 80)(5 114 32 67 17 103 50 76)(6 110 33 63 18 99 51 87)(7 106 34 74 19 95 52 83)(8 117 35 70 20 91 53 79)(9 113 36 66 21 102 54 90)(10 109 37 62 22 98 55 86)(11 120 38 73 23 94 56 82)(12 116 39 69 24 105 57 78)(13 112 40 65 25 101 58 89)(14 108 41 61 26 97 59 85)(15 119 42 72 27 93 60 81)
(1 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 46)(44 47)(45 48)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F5A5B6A6B8A8B8C8D8E8F8G8H10A10B10C10D12A12B12C15A15B20A20B20C20D20E20F24A24B24C24D30A···30F40A···40H60A···60H
order12222234444445566888888881010101012121215152020202020202424242430···3040···4060···60
size11215153021121515302224666610101010224422444222244202020204···412···124···4

66 irreducible representations

dim111111111222222222224444444
type++++++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D5D6D6M4(2)D10D10C4×S3C4×S3C4×D5C4×D5S3×D5S3×M4(2)D30.C2C2×S3×D5D30.C2D5×M4(2)D154M4(2)
kernelD154M4(2)D152C8D30.5C4C3×C4.Dic5C5×C4.Dic3C2×C4×D15C4×D15C2×Dic15C22×D15C4.Dic5C4.Dic3C52C8C2×C20D15C3⋊C8C2×C12C20C2×C10C12C2×C6C2×C4C5C4C4C22C3C1
# reps122111422122144222442222248

Matrix representation of D154M4(2) in GL6(𝔽241)

0510000
1891890000
00240100
00240000
000010
000001
,
178840000
79630000
00240000
00240100
000010
000001
,
6400000
0640000
00017700
00177000
00000240
0000640
,
24000000
02400000
001000
000100
00002400
000001

G:=sub<GL(6,GF(241))| [0,189,0,0,0,0,51,189,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[178,79,0,0,0,0,84,63,0,0,0,0,0,0,240,240,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,0,177,0,0,0,0,177,0,0,0,0,0,0,0,0,64,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,1] >;

D154M4(2) in GAP, Magma, Sage, TeX

D_{15}\rtimes_4M_4(2)
% in TeX

G:=Group("D15:4M4(2)");
// GroupNames label

G:=SmallGroup(480,368);
// by ID

G=gap.SmallGroup(480,368);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,219,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^8=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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