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## G = C3×C23.1D10order 480 = 25·3·5

### Direct product of C3 and C23.1D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C23.1D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×C30 — C6×C5⋊D4 — C3×C23.1D10
 Lower central C5 — C10 — C2×C10 — C3×C23.1D10
 Upper central C1 — C6 — C22×C6 — C3×C22⋊C4

Generators and relations for C3×C23.1D10
G = < a,b,c,d,e | a3=b2=c2=d20=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bd-1 >

Subgroups: 416 in 104 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4 [×3], C22 [×3], C22 [×3], C5, C6, C6 [×4], C2×C4 [×3], D4 [×2], C23, C23, D5, C10, C10 [×3], C12 [×3], C2×C6 [×3], C2×C6 [×3], C15, C22⋊C4, C22⋊C4, C2×D4, Dic5 [×2], C20, D10 [×2], C2×C10 [×3], C2×C10, C2×C12 [×3], C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×3], C23⋊C4, C2×Dic5, C2×Dic5, C5⋊D4 [×2], C2×C20, C22×D5, C22×C10, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C3×Dic5 [×2], C60, C6×D5 [×2], C2×C30 [×3], C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C3×C23⋊C4, C6×Dic5, C6×Dic5, C3×C5⋊D4 [×2], C2×C60, D5×C2×C6, C22×C30, C23.1D10, C3×C23.D5, C15×C22⋊C4, C6×C5⋊D4, C3×C23.1D10
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D10, C2×C12, C3×D4 [×2], C3×D5, C23⋊C4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C23⋊C4, D5×C12, C3×D20, C3×C5⋊D4, C23.1D10, C3×D10⋊C4, C3×C23.1D10

Smallest permutation representation of C3×C23.1D10
On 120 points
Generators in S120
(1 25 120)(2 26 101)(3 27 102)(4 28 103)(5 29 104)(6 30 105)(7 31 106)(8 32 107)(9 33 108)(10 34 109)(11 35 110)(12 36 111)(13 37 112)(14 38 113)(15 39 114)(16 40 115)(17 21 116)(18 22 117)(19 23 118)(20 24 119)(41 83 72)(42 84 73)(43 85 74)(44 86 75)(45 87 76)(46 88 77)(47 89 78)(48 90 79)(49 91 80)(50 92 61)(51 93 62)(52 94 63)(53 95 64)(54 96 65)(55 97 66)(56 98 67)(57 99 68)(58 100 69)(59 81 70)(60 82 71)
(1 70)(3 72)(5 74)(7 76)(9 78)(11 80)(13 62)(15 64)(17 66)(19 68)(21 55)(23 57)(25 59)(27 41)(29 43)(31 45)(33 47)(35 49)(37 51)(39 53)(81 120)(83 102)(85 104)(87 106)(89 108)(91 110)(93 112)(95 114)(97 116)(99 118)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(81 120)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)(100 119)
(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 64 70 15)(2 63)(3 13 72 62)(4 12)(5 80 74 11)(6 79)(7 9 76 78)(10 75)(14 71)(16 20)(17 68 66 19)(18 67)(21 57 55 23)(22 56)(24 40)(25 53 59 39)(26 52)(27 37 41 51)(28 36)(29 49 43 35)(30 48)(31 33 45 47)(34 44)(38 60)(42 50)(54 58)(61 73)(65 69)(81 114 120 95)(82 113)(83 93 102 112)(84 92)(85 110 104 91)(86 109)(87 89 106 108)(90 105)(94 101)(96 100)(97 118 116 99)(98 117)(103 111)(115 119)

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

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

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

93 conjugacy classes

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

93 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 4 4 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D4 D5 D10 C3×D4 C3×D5 C4×D5 D20 C5⋊D4 C6×D5 D5×C12 C3×D20 C3×C5⋊D4 C23⋊C4 C3×C23⋊C4 C23.1D10 C3×C23.1D10 kernel C3×C23.1D10 C3×C23.D5 C15×C22⋊C4 C6×C5⋊D4 C23.1D10 C6×Dic5 D5×C2×C6 C23.D5 C5×C22⋊C4 C2×C5⋊D4 C2×Dic5 C22×D5 C2×C30 C3×C22⋊C4 C22×C6 C2×C10 C22⋊C4 C2×C6 C2×C6 C2×C6 C23 C22 C22 C22 C15 C5 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 2 2 2 4 4 4 4 4 4 8 8 8 1 2 4 8

Matrix representation of C3×C23.1D10 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
,
 60 0 0 0 0 60 0 0 60 0 1 0 0 60 0 1
,
 60 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 43 60 36 2 1 0 59 0 56 45 18 1 16 39 60 0
,
 53 31 0 0 53 8 0 0 5 16 43 60 48 56 18 18
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[60,0,60,0,0,60,0,60,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[43,1,56,16,60,0,45,39,36,59,18,60,2,0,1,0],[53,53,5,48,31,8,16,56,0,0,43,18,0,0,60,18] >;

C3×C23.1D10 in GAP, Magma, Sage, TeX

C_3\times C_2^3._1D_{10}
% in TeX

G:=Group("C3xC2^3.1D10");
// GroupNames label

G:=SmallGroup(480,84);
// by ID

G=gap.SmallGroup(480,84);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,1683,1271,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^20=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations

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