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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 [×8], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], C5, S3 [×5], C6, C6 [×3], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], D5 [×5], C10, C10 [×3], Dic3 [×4], C12, C12 [×3], D6 [×10], C2×C6 [×3], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], Dic5 [×4], C20, C20 [×3], D10 [×10], C2×C10 [×3], Dic6 [×3], C4×S3 [×10], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], D15 [×2], D15 [×3], C30, C30 [×3], C2×C4○D4, Dic10 [×3], C4×D5 [×10], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, Dic15, Dic15 [×3], C60, C60 [×3], D30, D30 [×3], D30 [×6], C2×C30 [×3], C2×C4×D5 [×3], C4○D20 [×3], D4×D5 [×3], D42D5 [×3], Q8×D5, Q82D5, C5×C4○D4, S3×C4○D4, Dic30 [×3], C4×D15, C4×D15 [×9], D60 [×3], C2×Dic15 [×3], C157D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, C22×D15 [×3], D5×C4○D4, C2×C4×D15 [×3], D6011C2 [×3], D4×D15 [×3], D42D15 [×3], Q8×D15, Q83D15, C15×C4○D4, C4○D4×D15
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], D15, C2×C4○D4, C22×D5 [×7], S3×C23, D30 [×7], C23×D5, S3×C4○D4, C22×D15 [×7], D5×C4○D4, C23×D15, C4○D4×D15

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