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## G = C15×C4○D4order 240 = 24·3·5

### Direct product of C15 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C15×C4○D4
 Chief series C1 — C2 — C10 — C30 — C2×C30 — D4×C15 — C15×C4○D4
 Lower central C1 — C2 — C15×C4○D4
 Upper central C1 — C60 — C15×C4○D4

Generators and relations for C15×C4○D4
G = < a,b,c,d | a15=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 92 in 80 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, D4, Q8, C10, C10, C12, C12, C2×C6, C15, C4○D4, C20, C20, C2×C10, C2×C12, C3×D4, C3×Q8, C30, C30, C2×C20, C5×D4, C5×Q8, C3×C4○D4, C60, C60, C2×C30, C5×C4○D4, C2×C60, D4×C15, Q8×C15, C15×C4○D4
Quotients: C1, C2, C3, C22, C5, C6, C23, C10, C2×C6, C15, C4○D4, C2×C10, C22×C6, C30, C22×C10, C3×C4○D4, C2×C30, C5×C4○D4, C22×C30, C15×C4○D4

Smallest permutation representation of C15×C4○D4
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 75 33 29)(2 61 34 30)(3 62 35 16)(4 63 36 17)(5 64 37 18)(6 65 38 19)(7 66 39 20)(8 67 40 21)(9 68 41 22)(10 69 42 23)(11 70 43 24)(12 71 44 25)(13 72 45 26)(14 73 31 27)(15 74 32 28)(46 119 105 77)(47 120 91 78)(48 106 92 79)(49 107 93 80)(50 108 94 81)(51 109 95 82)(52 110 96 83)(53 111 97 84)(54 112 98 85)(55 113 99 86)(56 114 100 87)(57 115 101 88)(58 116 102 89)(59 117 103 90)(60 118 104 76)
(1 29 33 75)(2 30 34 61)(3 16 35 62)(4 17 36 63)(5 18 37 64)(6 19 38 65)(7 20 39 66)(8 21 40 67)(9 22 41 68)(10 23 42 69)(11 24 43 70)(12 25 44 71)(13 26 45 72)(14 27 31 73)(15 28 32 74)(46 119 105 77)(47 120 91 78)(48 106 92 79)(49 107 93 80)(50 108 94 81)(51 109 95 82)(52 110 96 83)(53 111 97 84)(54 112 98 85)(55 113 99 86)(56 114 100 87)(57 115 101 88)(58 116 102 89)(59 117 103 90)(60 118 104 76)
(1 59)(2 60)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 76)(31 101)(32 102)(33 103)(34 104)(35 105)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(61 118)(62 119)(63 120)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)

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

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

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

C15×C4○D4 is a maximal subgroup of   Q83Dic15  D4.Dic15  D4⋊D30  D4.8D30  D4.9D30  D48D30  D4.10D30
C15×C4○D4 is a maximal quotient of   D4×C60  Q8×C60

150 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 6C ··· 6H 10A 10B 10C 10D 10E ··· 10P 12A 12B 12C 12D 12E ··· 12J 15A ··· 15H 20A ··· 20H 20I ··· 20T 30A ··· 30H 30I ··· 30AF 60A ··· 60P 60Q ··· 60AN order 1 2 2 2 2 3 3 4 4 4 4 4 5 5 5 5 6 6 6 ··· 6 10 10 10 10 10 ··· 10 12 12 12 12 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 2 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C2 C3 C5 C6 C6 C6 C10 C10 C10 C15 C30 C30 C30 C4○D4 C3×C4○D4 C5×C4○D4 C15×C4○D4 kernel C15×C4○D4 C2×C60 D4×C15 Q8×C15 C5×C4○D4 C3×C4○D4 C2×C20 C5×D4 C5×Q8 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C15 C5 C3 C1 # reps 1 3 3 1 2 4 6 6 2 12 12 4 8 24 24 8 2 4 8 16

Matrix representation of C15×C4○D4 in GL3(𝔽61) generated by

 13 0 0 0 58 0 0 0 58
,
 1 0 0 0 50 0 0 0 50
,
 60 0 0 0 11 11 0 0 50
,
 60 0 0 0 11 11 0 39 50
G:=sub<GL(3,GF(61))| [13,0,0,0,58,0,0,0,58],[1,0,0,0,50,0,0,0,50],[60,0,0,0,11,0,0,11,50],[60,0,0,0,11,39,0,11,50] >;

C15×C4○D4 in GAP, Magma, Sage, TeX

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

G:=Group("C15xC4oD4");
// GroupNames label

G:=SmallGroup(240,188);
// by ID

G=gap.SmallGroup(240,188);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-2,1465,554]);
// Polycyclic

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

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