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## G = C3×C23.F5order 480 = 25·3·5

### Direct product of C3 and C23.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C23.F5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C6×Dic5 — C3×C22.F5 — C3×C23.F5
 Lower central C5 — C10 — C2×C10 — C3×C23.F5
 Upper central C1 — C6 — C2×C6 — C22×C6

Generators and relations for C3×C23.F5
G = < a,b,c,d,e,f | a3=b2=c2=d2=e5=1, f4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >

Subgroups: 376 in 92 conjugacy classes, 32 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, Dic5, D10, C2×C10, C2×C10, C24, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C4.D4, C5⋊C8, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C3×M4(2), C6×D4, C3×Dic5, C6×D5, C2×C30, C2×C30, C22.F5, C2×C5⋊D4, C3×C4.D4, C3×C5⋊C8, C6×Dic5, C3×C5⋊D4, D5×C2×C6, C22×C30, C23.F5, C3×C22.F5, C6×C5⋊D4, C3×C23.F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, F5, C2×C12, C3×D4, C4.D4, C2×F5, C3×C22⋊C4, C3×F5, C22⋊F5, C3×C4.D4, C6×F5, C23.F5, C3×C22⋊F5, C3×C23.F5

Smallest permutation representation of C3×C23.F5
On 120 points
Generators in S120
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 46)(10 87 47)(11 88 48)(12 81 41)(13 82 42)(14 83 43)(15 84 44)(16 85 45)(17 90 115)(18 91 116)(19 92 117)(20 93 118)(21 94 119)(22 95 120)(23 96 113)(24 89 114)(25 105 66)(26 106 67)(27 107 68)(28 108 69)(29 109 70)(30 110 71)(31 111 72)(32 112 65)(33 53 74)(34 54 75)(35 55 76)(36 56 77)(37 49 78)(38 50 79)(39 51 80)(40 52 73)
(2 6)(3 7)(10 14)(11 15)(17 21)(20 24)(26 30)(27 31)(35 39)(36 40)(43 47)(44 48)(51 55)(52 56)(57 61)(58 62)(67 71)(68 72)(73 77)(76 80)(83 87)(84 88)(89 93)(90 94)(97 101)(98 102)(106 110)(107 111)(114 118)(115 119)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(81 85)(83 87)(89 93)(91 95)(97 101)(99 103)(106 110)(108 112)(114 118)(116 120)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)
(1 54 46 19 105)(2 20 55 106 47)(3 107 21 48 56)(4 41 108 49 22)(5 50 42 23 109)(6 24 51 110 43)(7 111 17 44 52)(8 45 112 53 18)(9 92 66 100 75)(10 101 93 76 67)(11 77 102 68 94)(12 69 78 95 103)(13 96 70 104 79)(14 97 89 80 71)(15 73 98 72 90)(16 65 74 91 99)(25 60 34 86 117)(26 87 61 118 35)(27 119 88 36 62)(28 37 120 63 81)(29 64 38 82 113)(30 83 57 114 39)(31 115 84 40 58)(32 33 116 59 85)
(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)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 5 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 10A ··· 10G 12A 12B 12C 12D 15A 15B 24A ··· 24H 30A ··· 30N order 1 2 2 2 2 3 3 4 4 5 6 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 15 15 24 ··· 24 30 ··· 30 size 1 1 2 4 20 1 1 10 10 4 1 1 2 2 4 4 20 20 20 20 20 20 4 ··· 4 10 10 10 10 4 4 20 ··· 20 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 C3×D4 F5 C4.D4 C2×F5 C3×F5 C22⋊F5 C3×C4.D4 C6×F5 C23.F5 C3×C22⋊F5 C3×C23.F5 kernel C3×C23.F5 C3×C22.F5 C6×C5⋊D4 C23.F5 D5×C2×C6 C22×C30 C22.F5 C2×C5⋊D4 C22×D5 C22×C10 C3×Dic5 Dic5 C22×C6 C15 C2×C6 C23 C6 C5 C22 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 1 1 1 2 2 2 2 4 4 8

Matrix representation of C3×C23.F5 in GL8(𝔽241)

 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 214 1 0 0 0 0 0 222 0 0 240
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 237 27 240 0 0 0 0 0 222 179 0 240
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240
,
 240 1 0 0 0 0 0 0 240 0 1 0 0 0 0 0 240 0 0 1 0 0 0 0 240 0 0 0 0 0 0 0 0 0 0 0 91 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 233 28 87 0 0 0 0 0 119 184 0 205
,
 125 57 88 215 0 0 0 0 213 31 85 99 0 0 0 0 210 156 142 187 0 0 0 0 26 3 116 184 0 0 0 0 0 0 0 0 237 27 239 0 0 0 0 0 222 179 0 239 0 0 0 0 113 193 4 214 0 0 0 0 25 99 19 62

G:=sub<GL(8,GF(241))| [15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,222,0,0,0,0,0,240,214,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,237,222,0,0,0,0,0,1,27,179,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,91,0,233,119,0,0,0,0,0,98,28,184,0,0,0,0,0,0,87,0,0,0,0,0,0,0,0,205],[125,213,210,26,0,0,0,0,57,31,156,3,0,0,0,0,88,85,142,116,0,0,0,0,215,99,187,184,0,0,0,0,0,0,0,0,237,222,113,25,0,0,0,0,27,179,193,99,0,0,0,0,239,0,4,19,0,0,0,0,0,239,214,62] >;

C3×C23.F5 in GAP, Magma, Sage, TeX

C_3\times C_2^3.F_5
% in TeX

G:=Group("C3xC2^3.F5");
// GroupNames label

G:=SmallGroup(480,293);
// by ID

G=gap.SmallGroup(480,293);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,850,136,2524,9414,1595]);
// Polycyclic

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

׿
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