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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

Aliases: C15×C4○D4, D42C30, Q83C30, C30.60C23, C60.81C22, (C5×D4)C60, (C5×Q8)C60, (C2×C20)⋊7C6, (C2×C4)⋊3C30, C60(D4×C15), (C5×D4)⋊5C6, C60(Q8×C15), (C5×Q8)⋊7C6, (C2×C60)⋊15C2, (C2×C12)⋊7C10, (C3×D4)⋊5C10, C4.5(C2×C30), (C3×Q8)⋊5C10, C22.(C2×C30), (D4×C15)⋊11C2, C20.21(C2×C6), (Q8×C15)⋊11C2, C12.21(C2×C10), C2.3(C22×C30), (C2×C30).21C22, C6.13(C22×C10), C10.13(C22×C6), (C2×C10).2(C2×C6), (C2×C6).2(C2×C10), SmallGroup(240,188)

Series: Derived Chief Lower central Upper central

C1C2 — C15×C4○D4
C1C2C10C30C2×C30D4×C15 — C15×C4○D4
C1C2 — C15×C4○D4
C1C60 — 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 [×3], C3, C4, C4 [×3], C22 [×3], C5, C6, C6 [×3], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C12, C12 [×3], C2×C6 [×3], C15, C4○D4, C20, C20 [×3], C2×C10 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C30, C30 [×3], C2×C20 [×3], C5×D4 [×3], C5×Q8, C3×C4○D4, C60, C60 [×3], C2×C30 [×3], C5×C4○D4, C2×C60 [×3], D4×C15 [×3], Q8×C15, C15×C4○D4
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], C23, C10 [×7], C2×C6 [×7], C15, C4○D4, C2×C10 [×7], C22×C6, C30 [×7], C22×C10, C3×C4○D4, C2×C30 [×7], 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)
(1 108 50 104)(2 109 51 105)(3 110 52 91)(4 111 53 92)(5 112 54 93)(6 113 55 94)(7 114 56 95)(8 115 57 96)(9 116 58 97)(10 117 59 98)(11 118 60 99)(12 119 46 100)(13 120 47 101)(14 106 48 102)(15 107 49 103)(16 87 75 37)(17 88 61 38)(18 89 62 39)(19 90 63 40)(20 76 64 41)(21 77 65 42)(22 78 66 43)(23 79 67 44)(24 80 68 45)(25 81 69 31)(26 82 70 32)(27 83 71 33)(28 84 72 34)(29 85 73 35)(30 86 74 36)
(1 104 50 108)(2 105 51 109)(3 91 52 110)(4 92 53 111)(5 93 54 112)(6 94 55 113)(7 95 56 114)(8 96 57 115)(9 97 58 116)(10 98 59 117)(11 99 60 118)(12 100 46 119)(13 101 47 120)(14 102 48 106)(15 103 49 107)(16 87 75 37)(17 88 61 38)(18 89 62 39)(19 90 63 40)(20 76 64 41)(21 77 65 42)(22 78 66 43)(23 79 67 44)(24 80 68 45)(25 81 69 31)(26 82 70 32)(27 83 71 33)(28 84 72 34)(29 85 73 35)(30 86 74 36)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(31 104)(32 105)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 61)(58 62)(59 63)(60 64)(76 118)(77 119)(78 120)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)

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

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

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

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 2A2B2C2D3A3B4A4B4C4D4E5A5B5C5D6A6B6C···6H10A10B10C10D10E···10P12A12B12C12D12E···12J15A···15H20A···20H20I···20T30A···30H30I···30AF60A···60P60Q···60AN
order1222233444445555666···61010101010···101212121212···1215···1520···2020···2030···3030···3060···6060···60
size1122211112221111112···211112···211112···21···11···12···21···12···21···12···2

150 irreducible representations

dim11111111111111112222
type++++
imageC1C2C2C2C3C5C6C6C6C10C10C10C15C30C30C30C4○D4C3×C4○D4C5×C4○D4C15×C4○D4
kernelC15×C4○D4C2×C60D4×C15Q8×C15C5×C4○D4C3×C4○D4C2×C20C5×D4C5×Q8C2×C12C3×D4C3×Q8C4○D4C2×C4D4Q8C15C5C3C1
# reps1331246621212482424824816

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

1300
0580
0058
,
100
0500
0050
,
6000
01111
0050
,
6000
01111
03950
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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