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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 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23, C23, D5, C10, C10 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, Dic5 [×2], D10 [×2], C2×C10, C2×C10 [×2], C24 [×2], C2×C12, C3×D4 [×2], C22×C6, C22×C6, C3×D5, C30, C30 [×2], C4.D4, C5⋊C8 [×2], C2×Dic5, C5⋊D4 [×2], C22×D5, C22×C10, C3×M4(2) [×2], C6×D4, C3×Dic5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×2], C22.F5 [×2], C2×C5⋊D4, C3×C4.D4, C3×C5⋊C8 [×2], C6×Dic5, C3×C5⋊D4 [×2], D5×C2×C6, C22×C30, C23.F5, C3×C22.F5 [×2], C6×C5⋊D4, C3×C23.F5
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C12 [×2], C2×C6, C22⋊C4, F5, C2×C12, C3×D4 [×2], 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 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 94 119)(18 95 120)(19 96 113)(20 89 114)(21 90 115)(22 91 116)(23 92 117)(24 93 118)(25 45 66)(26 46 67)(27 47 68)(28 48 69)(29 41 70)(30 42 71)(31 43 72)(32 44 65)(33 108 74)(34 109 75)(35 110 76)(36 111 77)(37 112 78)(38 105 79)(39 106 80)(40 107 73)
(2 6)(3 7)(9 13)(12 16)(17 21)(20 24)(27 31)(28 32)(35 39)(36 40)(43 47)(44 48)(50 54)(51 55)(57 61)(58 62)(65 69)(68 72)(73 77)(76 80)(81 85)(82 86)(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)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(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)
(1 109 46 23 53)(2 24 110 54 47)(3 55 17 48 111)(4 41 56 112 18)(5 105 42 19 49)(6 20 106 50 43)(7 51 21 44 107)(8 45 52 108 22)(9 90 65 73 98)(10 74 91 99 66)(11 100 75 67 92)(12 68 101 93 76)(13 94 69 77 102)(14 78 95 103 70)(15 104 79 71 96)(16 72 97 89 80)(25 87 33 116 59)(26 117 88 60 34)(27 61 118 35 81)(28 36 62 82 119)(29 83 37 120 63)(30 113 84 64 38)(31 57 114 39 85)(32 40 58 86 115)
(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)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (2,6)(3,7)(9,13)(12,16)(17,21)(20,24)(27,31)(28,32)(35,39)(36,40)(43,47)(44,48)(50,54)(51,55)(57,61)(58,62)(65,69)(68,72)(73,77)(76,80)(81,85)(82,86)(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)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(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), (1,109,46,23,53)(2,24,110,54,47)(3,55,17,48,111)(4,41,56,112,18)(5,105,42,19,49)(6,20,106,50,43)(7,51,21,44,107)(8,45,52,108,22)(9,90,65,73,98)(10,74,91,99,66)(11,100,75,67,92)(12,68,101,93,76)(13,94,69,77,102)(14,78,95,103,70)(15,104,79,71,96)(16,72,97,89,80)(25,87,33,116,59)(26,117,88,60,34)(27,61,118,35,81)(28,36,62,82,119)(29,83,37,120,63)(30,113,84,64,38)(31,57,114,39,85)(32,40,58,86,115), (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)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (2,6)(3,7)(9,13)(12,16)(17,21)(20,24)(27,31)(28,32)(35,39)(36,40)(43,47)(44,48)(50,54)(51,55)(57,61)(58,62)(65,69)(68,72)(73,77)(76,80)(81,85)(82,86)(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)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(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), (1,109,46,23,53)(2,24,110,54,47)(3,55,17,48,111)(4,41,56,112,18)(5,105,42,19,49)(6,20,106,50,43)(7,51,21,44,107)(8,45,52,108,22)(9,90,65,73,98)(10,74,91,99,66)(11,100,75,67,92)(12,68,101,93,76)(13,94,69,77,102)(14,78,95,103,70)(15,104,79,71,96)(16,72,97,89,80)(25,87,33,116,59)(26,117,88,60,34)(27,61,118,35,81)(28,36,62,82,119)(29,83,37,120,63)(30,113,84,64,38)(31,57,114,39,85)(32,40,58,86,115), (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),(9,86,51),(10,87,52),(11,88,53),(12,81,54),(13,82,55),(14,83,56),(15,84,49),(16,85,50),(17,94,119),(18,95,120),(19,96,113),(20,89,114),(21,90,115),(22,91,116),(23,92,117),(24,93,118),(25,45,66),(26,46,67),(27,47,68),(28,48,69),(29,41,70),(30,42,71),(31,43,72),(32,44,65),(33,108,74),(34,109,75),(35,110,76),(36,111,77),(37,112,78),(38,105,79),(39,106,80),(40,107,73)], [(2,6),(3,7),(9,13),(12,16),(17,21),(20,24),(27,31),(28,32),(35,39),(36,40),(43,47),(44,48),(50,54),(51,55),(57,61),(58,62),(65,69),(68,72),(73,77),(76,80),(81,85),(82,86),(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),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(50,54),(52,56),(57,61),(59,63),(66,70),(68,72),(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)], [(1,109,46,23,53),(2,24,110,54,47),(3,55,17,48,111),(4,41,56,112,18),(5,105,42,19,49),(6,20,106,50,43),(7,51,21,44,107),(8,45,52,108,22),(9,90,65,73,98),(10,74,91,99,66),(11,100,75,67,92),(12,68,101,93,76),(13,94,69,77,102),(14,78,95,103,70),(15,104,79,71,96),(16,72,97,89,80),(25,87,33,116,59),(26,117,88,60,34),(27,61,118,35,81),(28,36,62,82,119),(29,83,37,120,63),(30,113,84,64,38),(31,57,114,39,85),(32,40,58,86,115)], [(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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