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## G = C4○D4×D15order 480 = 25·3·5

### Direct product of C4○D4 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C4○D4×D15
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C2×C4×D15 — C4○D4×D15
 Lower central C15 — C30 — C4○D4×D15
 Upper central C1 — C4 — C4○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 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 6D 10A 10B 10C ··· 10H 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A 20B 20C 20D 20E ··· 20J 30A 30B 30C 30D 30E ··· 30P 60A ··· 60H 60I ··· 60T order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 10 10 10 ··· 10 12 12 12 12 12 15 15 15 15 20 20 20 20 20 ··· 20 30 30 30 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 2 2 15 15 30 30 30 2 1 1 2 2 2 15 15 30 30 30 2 2 2 4 4 4 2 2 4 ··· 4 2 2 4 4 4 2 2 2 2 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 D15 D30 D30 D30 S3×C4○D4 D5×C4○D4 C4○D4×D15 kernel C4○D4×D15 C2×C4×D15 D60⋊11C2 D4×D15 D4⋊2D15 Q8×D15 Q8⋊3D15 C15×C4○D4 C5×C4○D4 C3×C4○D4 C2×C20 C5×D4 C5×Q8 D15 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C5 C3 C1 # reps 1 3 3 3 3 1 1 1 1 2 3 3 1 4 6 6 2 4 12 12 4 2 4 8

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

 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 9 0 0 0 0 54 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 0 0 0 0 0 7 60
,
 60 43 0 0 0 0 18 18 0 0 0 0 0 0 1 20 0 0 0 0 9 59 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 43 0 0 0 0 0 1 0 0 0 0 0 0 1 20 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60

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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