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G = C3×C23.F5order 480 = 25·3·5

Direct product of C3 and C23.F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C3×C23.F5, C23.(C3×F5), C22.F51C6, C22.5(C6×F5), (C22×C6).1F5, C156(C4.D4), (C22×C30).9C4, Dic5.4(C3×D4), (C22×C10).6C12, (C3×Dic5).43D4, (C22×D5).2C12, C6.37(C22⋊F5), C30.37(C22⋊C4), (C6×Dic5).172C22, C5⋊(C3×C4.D4), (D5×C2×C6).4C4, (C2×C5⋊D4).8C6, (C2×C6).29(C2×F5), (C2×C30).55(C2×C4), (C6×C5⋊D4).17C2, (C3×C22.F5)⋊5C2, (C2×C10).12(C2×C12), C2.11(C3×C22⋊F5), C10.11(C3×C22⋊C4), (C2×Dic5).21(C2×C6), SmallGroup(480,293)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C23.F5
C1C5C10C2×C10C2×Dic5C6×Dic5C3×C22.F5 — C3×C23.F5
C5C10C2×C10 — C3×C23.F5
C1C6C2×C6C22×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 2A2B2C2D3A3B4A4B 5 6A6B6C6D6E6F6G6H8A8B8C8D10A···10G12A12B12C12D15A15B24A···24H30A···30N
order122223344566666666888810···1012121212151524···2430···30
size11242011101041122442020202020204···4101010104420···204···4

57 irreducible representations

dim1111111111224444444444
type++++++++
imageC1C2C2C3C4C4C6C6C12C12D4C3×D4F5C4.D4C2×F5C3×F5C22⋊F5C3×C4.D4C6×F5C23.F5C3×C22⋊F5C3×C23.F5
kernelC3×C23.F5C3×C22.F5C6×C5⋊D4C23.F5D5×C2×C6C22×C30C22.F5C2×C5⋊D4C22×D5C22×C10C3×Dic5Dic5C22×C6C15C2×C6C23C6C5C22C3C2C1
# reps1212224244241112222448

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

150000000
015000000
001500000
000150000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
0000024000
0000021410
000022200240
,
10000000
01000000
00100000
00010000
00001000
00000100
0000237272400
00002221790240
,
10000000
01000000
00100000
00010000
0000240000
0000024000
0000002400
0000000240
,
2401000000
2400100000
2400010000
2400000000
000091000
000009800
000023328870
00001191840205
,
12557882150000
2133185990000
2101561421870000
2631161840000
0000237272390
00002221790239
00001131934214
000025991962

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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