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G = C5⋊C8⋊D6order 480 = 25·3·5

1st semidirect product of C5⋊C8 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5⋊C85D6, D5⋊C84S3, D5⋊(C8⋊S3), (C3×D5)⋊M4(2), D6.F57C2, D6.8(C2×F5), (C4×S3).4F5, C4.32(S3×F5), (S3×C20).6C4, C20.32(C4×S3), C60.32(C2×C4), (C4×D15).6C4, (C4×D5).74D6, C153(C2×M4(2)), C12.39(C2×F5), C60.C45C2, C15⋊C86C22, D10.23(C4×S3), D30.10(C2×C4), C31(D5⋊M4(2)), Dic3.F57C2, Dic3.9(C2×F5), (D5×Dic3).7C4, C6.12(C22×F5), C30.12(C22×C4), Dic15.12(C2×C4), (D5×C12).66C22, D30.C2.15C22, (C3×Dic5).30C23, (S3×Dic5).15C22, Dic5.32(C22×S3), C51(C2×C8⋊S3), (C2×S3×D5).7C4, (C3×D5⋊C8)⋊5C2, C2.16(C2×S3×F5), (C3×C5⋊C8)⋊6C22, C10.12(S3×C2×C4), (C4×S3×D5).11C2, (C6×D5).21(C2×C4), (S3×C10).10(C2×C4), (C5×Dic3).12(C2×C4), SmallGroup(480,993)

Series: Derived Chief Lower central Upper central

C1C30 — C5⋊C8⋊D6
C1C5C15C30C3×Dic5C3×C5⋊C8D6.F5 — C5⋊C8⋊D6
C15C30 — C5⋊C8⋊D6
C1C4

Generators and relations for C5⋊C8⋊D6
 G = < a,b,c,d | a5=b8=c6=d2=1, bab-1=a3, cac-1=dad=a-1, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 676 in 136 conjugacy classes, 50 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D5⋊C8, D5⋊C8, C4.F5, C22.F5, C2×C4×D5, C2×C8⋊S3, C3×C5⋊C8, C15⋊C8, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, D5⋊M4(2), D6.F5, Dic3.F5, C3×D5⋊C8, C60.C4, C4×S3×D5, C5⋊C8⋊D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, F5, C4×S3, C22×S3, C2×M4(2), C2×F5, C8⋊S3, S3×C2×C4, C22×F5, C2×C8⋊S3, S3×F5, D5⋊M4(2), C2×S3×F5, C5⋊C8⋊D6

Smallest permutation representation of C5⋊C8⋊D6
On 120 points
Generators in S120
(1 45 115 38 88)(2 39 46 81 116)(3 82 40 117 47)(4 118 83 48 33)(5 41 119 34 84)(6 35 42 85 120)(7 86 36 113 43)(8 114 87 44 37)(9 58 22 72 53)(10 65 59 54 23)(11 55 66 24 60)(12 17 56 61 67)(13 62 18 68 49)(14 69 63 50 19)(15 51 70 20 64)(16 21 52 57 71)(25 100 106 74 95)(26 75 101 96 107)(27 89 76 108 102)(28 109 90 103 77)(29 104 110 78 91)(30 79 97 92 111)(31 93 80 112 98)(32 105 94 99 73)
(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 58 99 5 62 103)(2 59 100 6 63 104)(3 60 101 7 64 97)(4 61 102 8 57 98)(9 73 84 18 90 45)(10 74 85 19 91 46)(11 75 86 20 92 47)(12 76 87 21 93 48)(13 77 88 22 94 41)(14 78 81 23 95 42)(15 79 82 24 96 43)(16 80 83 17 89 44)(25 35 69 110 116 54)(26 36 70 111 117 55)(27 37 71 112 118 56)(28 38 72 105 119 49)(29 39 65 106 120 50)(30 40 66 107 113 51)(31 33 67 108 114 52)(32 34 68 109 115 53)
(1 103)(2 100)(3 97)(4 102)(5 99)(6 104)(7 101)(8 98)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)(25 39)(26 36)(27 33)(28 38)(29 35)(30 40)(31 37)(32 34)(41 94)(42 91)(43 96)(44 93)(45 90)(46 95)(47 92)(48 89)(49 72)(50 69)(51 66)(52 71)(53 68)(54 65)(55 70)(56 67)(58 62)(60 64)(73 84)(74 81)(75 86)(76 83)(77 88)(78 85)(79 82)(80 87)(105 119)(106 116)(107 113)(108 118)(109 115)(110 120)(111 117)(112 114)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F 5 6A6B6C8A8B8C8D8E8F8G8H10A10B10C12A12B12C12D 15 20A20B20C20D24A···24H 30 60A60B
order122222344444456668888888810101012121212152020202024···24306060
size115563021155630421010101010103030303041212221010844121210···10888

48 irreducible representations

dim1111111111222222244444888
type+++++++++++++++
imageC1C2C2C2C2C2C4C4C4C4S3D6D6M4(2)C4×S3C4×S3C8⋊S3F5C2×F5C2×F5C2×F5D5⋊M4(2)S3×F5C2×S3×F5C5⋊C8⋊D6
kernelC5⋊C8⋊D6D6.F5Dic3.F5C3×D5⋊C8C60.C4C4×S3×D5D5×Dic3S3×C20C4×D15C2×S3×D5D5⋊C8C5⋊C8C4×D5C3×D5C20D10D5C4×S3Dic3C12D6C3C4C2C1
# reps1221112222121422811114112

Matrix representation of C5⋊C8⋊D6 in GL6(𝔽241)

100000
010000
0019024000
0019124000
001951955252
001770189240
,
24000000
02400000
000017764
0052111346
00100392400
0049382400
,
010000
24010000
0019024000
001905100
00002400
0019564521
,
100000
12400000
0019024000
001905100
00344710
0080224189240

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,190,191,195,177,0,0,240,240,195,0,0,0,0,0,52,189,0,0,0,0,52,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,52,100,49,0,0,0,1,39,38,0,0,177,113,240,240,0,0,64,46,0,0],[0,240,0,0,0,0,1,1,0,0,0,0,0,0,190,190,0,195,0,0,240,51,0,64,0,0,0,0,240,52,0,0,0,0,0,1],[1,1,0,0,0,0,0,240,0,0,0,0,0,0,190,190,34,80,0,0,240,51,47,224,0,0,0,0,1,189,0,0,0,0,0,240] >;

C5⋊C8⋊D6 in GAP, Magma, Sage, TeX

C_5\rtimes C_8\rtimes D_6
% in TeX

G:=Group("C5:C8:D6");
// GroupNames label

G:=SmallGroup(480,993);
// by ID

G=gap.SmallGroup(480,993);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,100,80,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^6=d^2=1,b*a*b^-1=a^3,c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

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