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## G = C5×Q8⋊3Dic3order 480 = 25·3·5

### Direct product of C5 and Q8⋊3Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×Q8⋊3Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C5×C4.Dic3 — C5×Q8⋊3Dic3
 Lower central C3 — C6 — C12 — C5×Q8⋊3Dic3
 Upper central C1 — C20 — C2×C20 — C5×C4○D4

Generators and relations for C5×Q83Dic3
G = < a,b,c,d,e | a5=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 180 in 88 conjugacy classes, 42 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C42, M4(2), C4○D4, C20, C20, C2×C10, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C30, C30, C4≀C2, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic3, C4×Dic3, C3×C4○D4, C5×Dic3, C60, C60, C2×C30, C2×C30, C4×C20, C5×M4(2), C5×C4○D4, Q83Dic3, C5×C3⋊C8, C10×Dic3, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C5×C4≀C2, C5×C4.Dic3, Dic3×C20, C15×C4○D4, C5×Q83Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, Dic3, D6, C22⋊C4, C20, C2×C10, C2×Dic3, C3⋊D4, C5×S3, C4≀C2, C2×C20, C5×D4, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, Q83Dic3, C10×Dic3, C5×C3⋊D4, C5×C4≀C2, C5×C6.D4, C5×Q83Dic3

Smallest permutation representation of C5×Q83Dic3
On 120 points
Generators in S120
(1 49 37 25 13)(2 50 38 26 14)(3 51 39 27 15)(4 52 40 28 16)(5 53 41 29 17)(6 54 42 30 18)(7 55 43 31 19)(8 56 44 32 20)(9 57 45 33 21)(10 58 46 34 22)(11 59 47 35 23)(12 60 48 36 24)(61 109 97 85 73)(62 110 98 86 74)(63 111 99 87 75)(64 112 100 88 76)(65 113 101 89 77)(66 114 102 90 78)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)(25 34 28 31)(26 35 29 32)(27 36 30 33)(37 46 40 43)(38 47 41 44)(39 48 42 45)(49 58 52 55)(50 59 53 56)(51 60 54 57)(61 67 64 70)(62 68 65 71)(63 69 66 72)(73 79 76 82)(74 80 77 83)(75 81 78 84)(85 91 88 94)(86 92 89 95)(87 93 90 96)(97 103 100 106)(98 104 101 107)(99 105 102 108)(109 115 112 118)(110 116 113 119)(111 117 114 120)
(1 67 4 70)(2 71 5 68)(3 69 6 72)(7 64 10 61)(8 62 11 65)(9 66 12 63)(13 79 16 82)(14 83 17 80)(15 81 18 84)(19 76 22 73)(20 74 23 77)(21 78 24 75)(25 91 28 94)(26 95 29 92)(27 93 30 96)(31 88 34 85)(32 86 35 89)(33 90 36 87)(37 103 40 106)(38 107 41 104)(39 105 42 108)(43 100 46 97)(44 98 47 101)(45 102 48 99)(49 115 52 118)(50 119 53 116)(51 117 54 120)(55 112 58 109)(56 110 59 113)(57 114 60 111)
(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 4)(2 6)(3 5)(7 10)(8 12)(9 11)(13 16)(14 18)(15 17)(19 22)(20 24)(21 23)(25 28)(26 30)(27 29)(31 34)(32 36)(33 35)(37 40)(38 42)(39 41)(43 46)(44 48)(45 47)(49 52)(50 54)(51 53)(55 58)(56 60)(57 59)(61 67 64 70)(62 72 65 69)(63 71 66 68)(73 79 76 82)(74 84 77 81)(75 83 78 80)(85 91 88 94)(86 96 89 93)(87 95 90 92)(97 103 100 106)(98 108 101 105)(99 107 102 104)(109 115 112 118)(110 120 113 117)(111 119 114 116)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 12C 12D 12E 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M 20N 20O 20P 20Q ··· 20AF 30A 30B 30C 30D 30E ··· 30P 40A ··· 40H 60A ··· 60H 60I ··· 60T order 1 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 12 12 15 15 15 15 20 ··· 20 20 20 20 20 20 20 20 20 20 ··· 20 30 30 30 30 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 size 1 1 2 4 2 1 1 2 4 6 6 6 6 1 1 1 1 2 4 4 4 12 12 1 1 1 1 2 2 2 2 4 4 4 4 2 2 4 4 4 2 2 2 2 1 ··· 1 2 2 2 2 4 4 4 4 6 ··· 6 2 2 2 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - - image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 S3 D4 D4 D6 Dic3 Dic3 C3⋊D4 C3⋊D4 C5×S3 C4≀C2 C5×D4 C5×D4 S3×C10 C5×Dic3 C5×Dic3 C5×C3⋊D4 C5×C3⋊D4 C5×C4≀C2 Q8⋊3Dic3 C5×Q8⋊3Dic3 kernel C5×Q8⋊3Dic3 C5×C4.Dic3 Dic3×C20 C15×C4○D4 D4×C15 Q8×C15 Q8⋊3Dic3 C4.Dic3 C4×Dic3 C3×C4○D4 C3×D4 C3×Q8 C5×C4○D4 C60 C2×C30 C2×C20 C5×D4 C5×Q8 C20 C2×C10 C4○D4 C15 C12 C2×C6 C2×C4 D4 Q8 C4 C22 C3 C5 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 8 8 16 2 8

Matrix representation of C5×Q83Dic3 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 64 0 0 0 43 177
,
 70 140 0 0 101 171 0 0 0 0 185 49 0 0 236 56
,
 240 240 0 0 1 0 0 0 0 0 1 0 0 0 140 240
,
 1 0 0 0 240 240 0 0 0 0 240 0 0 0 72 177
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,64,43,0,0,0,177],[70,101,0,0,140,171,0,0,0,0,185,236,0,0,49,56],[240,1,0,0,240,0,0,0,0,0,1,140,0,0,0,240],[1,240,0,0,0,240,0,0,0,0,240,72,0,0,0,177] >;

C5×Q83Dic3 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C5xQ8:3Dic3");
// GroupNames label

G:=SmallGroup(480,156);
// by ID

G=gap.SmallGroup(480,156);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,136,4204,2111,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations

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