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G = C4○D4×D15order 480 = 25·3·5

Direct product of C4○D4 and D15

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D4×D15, D47D30, Q87D30, D6027C22, C30.64C24, C60.88C23, D30.29C23, Dic3025C22, Dic15.31C23, (C2×C4)⋊7D30, (C5×D4)⋊23D6, (C2×C20)⋊13D6, (C5×Q8)⋊23D6, (C3×D4)⋊23D10, (D4×D15)⋊13C2, (C2×C12)⋊13D10, (C2×C60)⋊9C22, (C3×Q8)⋊20D10, (Q8×D15)⋊13C2, Q83D1513C2, D42D1513C2, C6.64(C23×D5), (D4×C15)⋊25C22, (C4×D15)⋊19C22, C157D411C22, (C2×C30).10C23, C10.64(S3×C23), D6011C217C2, (Q8×C15)⋊22C22, C4.43(C22×D15), C2.12(C23×D15), C20.138(C22×S3), C12.136(C22×D5), C22.2(C22×D15), (C2×Dic15)⋊27C22, (C22×D15).93C22, C57(S3×C4○D4), C37(D5×C4○D4), (C2×C4×D15)⋊6C2, (C5×C4○D4)⋊7S3, (C3×C4○D4)⋊3D5, C1529(C2×C4○D4), (C15×C4○D4)⋊3C2, (C2×C6).17(C22×D5), (C2×C10).18(C22×S3), SmallGroup(480,1175)

Series: Derived Chief Lower central Upper central

C1C30 — C4○D4×D15
C1C5C15C30D30C22×D15C2×C4×D15 — C4○D4×D15
C15C30 — C4○D4×D15
C1C4C4○D4

Generators and relations for C4○D4×D15
 G = < a,b,c,d,e | a4=c2=d15=e2=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1796 in 328 conjugacy classes, 121 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, D15, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, S3×C2×C4, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C3×C4○D4, Dic15, Dic15, C60, C60, D30, D30, D30, C2×C30, C2×C4×D5, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, S3×C4○D4, Dic30, C4×D15, C4×D15, D60, C2×Dic15, C157D4, C2×C60, D4×C15, Q8×C15, C22×D15, D5×C4○D4, C2×C4×D15, D6011C2, D4×D15, D42D15, Q8×D15, Q83D15, C15×C4○D4, C4○D4×D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, D15, C2×C4○D4, C22×D5, S3×C23, D30, C23×D5, S3×C4○D4, C22×D15, D5×C4○D4, C23×D15, C4○D4×D15

Smallest permutation representation of C4○D4×D15
On 120 points
Generators in S120
(1 54 17 39)(2 55 18 40)(3 56 19 41)(4 57 20 42)(5 58 21 43)(6 59 22 44)(7 60 23 45)(8 46 24 31)(9 47 25 32)(10 48 26 33)(11 49 27 34)(12 50 28 35)(13 51 29 36)(14 52 30 37)(15 53 16 38)(61 118 77 95)(62 119 78 96)(63 120 79 97)(64 106 80 98)(65 107 81 99)(66 108 82 100)(67 109 83 101)(68 110 84 102)(69 111 85 103)(70 112 86 104)(71 113 87 105)(72 114 88 91)(73 115 89 92)(74 116 90 93)(75 117 76 94)
(1 61 17 77)(2 62 18 78)(3 63 19 79)(4 64 20 80)(5 65 21 81)(6 66 22 82)(7 67 23 83)(8 68 24 84)(9 69 25 85)(10 70 26 86)(11 71 27 87)(12 72 28 88)(13 73 29 89)(14 74 30 90)(15 75 16 76)(31 102 46 110)(32 103 47 111)(33 104 48 112)(34 105 49 113)(35 91 50 114)(36 92 51 115)(37 93 52 116)(38 94 53 117)(39 95 54 118)(40 96 55 119)(41 97 56 120)(42 98 57 106)(43 99 58 107)(44 100 59 108)(45 101 60 109)
(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 89)(74 90)(75 76)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 106)(99 107)(100 108)(101 109)(102 110)(103 111)(104 112)(105 113)
(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 16)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(31 46)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 76)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(91 106)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D10A10B10C···10H12A12B12C12D12E15A15B15C15D20A20B20C20D20E···20J30A30B30C30D30E···30P60A···60H60I···60T
order122222222234444444444556666101010···101212121212151515152020202020···203030303030···3060···6060···60
size1122215153030302112221515303030222444224···422444222222224···422224···42···24···4

90 irreducible representations

dim111111112222222222222444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D5D6D6D6C4○D4D10D10D10D15D30D30D30S3×C4○D4D5×C4○D4C4○D4×D15
kernelC4○D4×D15C2×C4×D15D6011C2D4×D15D42D15Q8×D15Q83D15C15×C4○D4C5×C4○D4C3×C4○D4C2×C20C5×D4C5×Q8D15C2×C12C3×D4C3×Q8C4○D4C2×C4D4Q8C5C3C1
# reps13333111123314662412124248

Matrix representation of C4○D4×D15 in GL6(𝔽61)

6000000
0600000
0060000
0006000
0000500
0000050
,
6000000
0600000
001000
000100
0000609
0000541
,
100000
010000
001000
000100
000010
0000760
,
60430000
18180000
0012000
0095900
000010
000001
,
60430000
010000
0012000
0006000
0000600
0000060

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,54,0,0,0,0,9,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,7,0,0,0,0,0,60],[60,18,0,0,0,0,43,18,0,0,0,0,0,0,1,9,0,0,0,0,20,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,43,1,0,0,0,0,0,0,1,0,0,0,0,0,20,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

C4○D4×D15 in GAP, Magma, Sage, TeX

C_4\circ D_4\times D_{15}
% in TeX

G:=Group("C4oD4xD15");
// GroupNames label

G:=SmallGroup(480,1175);
// by ID

G=gap.SmallGroup(480,1175);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,346,2693,18822]);
// Polycyclic

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

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