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G = C12.F5order 240 = 24·3·5

1st non-split extension by C12 of F5 acting via F5/D5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C60.1C4, C12.1F5, C153M4(2), C20.1Dic3, Dic5.16D6, D10.3Dic3, C4.(C3⋊F5), C15⋊C85C2, C32(C4.F5), C6.9(C2×F5), C30.9(C2×C4), (C6×D5).5C4, (C4×D5).3S3, (D5×C12).5C2, C51(C4.Dic3), C10.2(C2×Dic3), (C3×Dic5).22C22, C2.4(C2×C3⋊F5), SmallGroup(240,119)

Series: Derived Chief Lower central Upper central

C1C30 — C12.F5
C1C5C15C30C3×Dic5C15⋊C8 — C12.F5
C15C30 — C12.F5
C1C2C4

Generators and relations for C12.F5
 G = < a,b,c | a12=b5=1, c4=a6, ab=ba, cac-1=a-1, cbc-1=b3 >

10C2
5C4
5C22
10C6
2D5
5C2×C4
15C8
15C8
5C2×C6
5C12
2C3×D5
15M4(2)
5C3⋊C8
5C3⋊C8
5C2×C12
3C5⋊C8
3C5⋊C8
5C4.Dic3
3C4.F5

Character table of C12.F5

 class 12A2B34A4B4C56A6B6C8A8B8C8D1012A12B12C12D15A15B20A20B30A30B60A60B60C60D
 size 111022554210103030303042210104444444444
ρ1111111111111111111111111111111    trivial
ρ211-11-11111-1-1-11-111-1-11111-1-111-1-1-1-1    linear of order 2
ρ311111111111-1-1-1-1111111111111111    linear of order 2
ρ411-11-11111-1-11-11-11-1-11111-1-111-1-1-1-1    linear of order 2
ρ511-111-1-111-1-1ii-i-i111-1-11111111111    linear of order 4
ρ61111-1-1-11111-iii-i1-1-1-1-111-1-111-1-1-1-1    linear of order 4
ρ71111-1-1-11111i-i-ii1-1-1-1-111-1-111-1-1-1-1    linear of order 4
ρ811-111-1-111-1-1-i-iii111-1-11111111111    linear of order 4
ρ9222-12222-1-1-100002-1-1-1-1-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ1022-2-1-2222-1110000211-1-1-1-1-2-2-1-11111    orthogonal lifted from D6
ρ1122-2-12-2-22-11100002-1-111-1-122-1-1-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ12222-1-2-2-22-1-1-1000021111-1-1-2-2-1-11111    symplectic lifted from Dic3, Schur index 2
ρ132-2020-2i2i2-2000000-200-2i2i2200-2-20000    complex lifted from M4(2)
ρ142-20202i-2i2-2000000-2002i-2i2200-2-20000    complex lifted from M4(2)
ρ152-20-10-2i2i21--3-30000-2-33i-i-1-100113-3-33    complex lifted from C4.Dic3
ρ162-20-102i-2i21--3-30000-23-3-ii-1-10011-333-3    complex lifted from C4.Dic3
ρ172-20-102i-2i21-3--30000-2-33-ii-1-100113-3-33    complex lifted from C4.Dic3
ρ182-20-10-2i2i21-3--30000-23-3i-i-1-10011-333-3    complex lifted from C4.Dic3
ρ194404400-14000000-14400-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ204404-400-14000000-1-4-400-1-111-1-11111    orthogonal lifted from C2×F5
ρ21440-2-400-1-2000000-122001+-15/21--15/2111--15/21+-15/2-1--15/2-1+-15/2-1--15/2-1+-15/2    complex lifted from C2×C3⋊F5
ρ22440-2400-1-2000000-1-2-2001--15/21+-15/2-1-11+-15/21--15/21--15/21+-15/21--15/21+-15/2    complex lifted from C3⋊F5
ρ23440-2400-1-2000000-1-2-2001+-15/21--15/2-1-11--15/21+-15/21+-15/21--15/21+-15/21--15/2    complex lifted from C3⋊F5
ρ24440-2-400-1-2000000-122001--15/21+-15/2111+-15/21--15/2-1+-15/2-1--15/2-1+-15/2-1--15/2    complex lifted from C2×C3⋊F5
ρ254-404000-1-400000010000-1-1-5--511-5-5--5--5    complex lifted from C4.F5
ρ264-404000-1-400000010000-1-1--5-511--5--5-5-5    complex lifted from C4.F5
ρ274-40-2000-12000000123-23001+-15/21--15/2-5--5-1+-15/2-1--15/24ζ34ζ534ζ5243ζ343ζ5443ζ54ζ324ζ544ζ543ζ3243ζ5343ζ52    complex faithful
ρ284-40-2000-120000001-2323001--15/21+-15/2-5--5-1--15/2-1+-15/243ζ343ζ5443ζ54ζ34ζ534ζ5243ζ3243ζ5343ζ524ζ324ζ544ζ5    complex faithful
ρ294-40-2000-120000001-2323001+-15/21--15/2--5-5-1+-15/2-1--15/24ζ324ζ544ζ543ζ3243ζ5343ζ524ζ34ζ534ζ5243ζ343ζ5443ζ5    complex faithful
ρ304-40-2000-12000000123-23001--15/21+-15/2--5-5-1--15/2-1+-15/243ζ3243ζ5343ζ524ζ324ζ544ζ543ζ343ζ5443ζ54ζ34ζ534ζ52    complex faithful

Smallest permutation representation of C12.F5
On 120 points
Generators in S120
(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 71 87 42 26)(2 72 88 43 27)(3 61 89 44 28)(4 62 90 45 29)(5 63 91 46 30)(6 64 92 47 31)(7 65 93 48 32)(8 66 94 37 33)(9 67 95 38 34)(10 68 96 39 35)(11 69 85 40 36)(12 70 86 41 25)(13 99 76 57 109)(14 100 77 58 110)(15 101 78 59 111)(16 102 79 60 112)(17 103 80 49 113)(18 104 81 50 114)(19 105 82 51 115)(20 106 83 52 116)(21 107 84 53 117)(22 108 73 54 118)(23 97 74 55 119)(24 98 75 56 120)
(1 116 10 119 7 110 4 113)(2 115 11 118 8 109 5 112)(3 114 12 117 9 120 6 111)(13 91 60 43 19 85 54 37)(14 90 49 42 20 96 55 48)(15 89 50 41 21 95 56 47)(16 88 51 40 22 94 57 46)(17 87 52 39 23 93 58 45)(18 86 53 38 24 92 59 44)(25 84 67 98 31 78 61 104)(26 83 68 97 32 77 62 103)(27 82 69 108 33 76 63 102)(28 81 70 107 34 75 64 101)(29 80 71 106 35 74 65 100)(30 79 72 105 36 73 66 99)

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

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

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

C12.F5 is a maximal subgroup of
D122F5  D605C4  D10.Dic6  D10.2Dic6  C40.Dic3  C24.1F5  Dic10⋊Dic3  D202Dic3  D12.2F5  S3×C4.F5  D60.C4  D15⋊M4(2)  C60.59(C2×C4)  Dic10.Dic3  D20.Dic3
C12.F5 is a maximal quotient of
C60⋊C8  C30.11C42  C30.7M4(2)

Matrix representation of C12.F5 in GL6(𝔽241)

400000
2381810000
00240000
00024000
00002400
00000240
,
100000
010000
00240240240240
001000
000100
000010
,
15790000
362260000
00240191191
00191191024
005074500
00217167167217

G:=sub<GL(6,GF(241))| [4,238,0,0,0,0,0,181,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[15,36,0,0,0,0,79,226,0,0,0,0,0,0,24,191,50,217,0,0,0,191,74,167,0,0,191,0,50,167,0,0,191,24,0,217] >;

C12.F5 in GAP, Magma, Sage, TeX

C_{12}.F_5
% in TeX

G:=Group("C12.F5");
// GroupNames label

G:=SmallGroup(240,119);
// by ID

G=gap.SmallGroup(240,119);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,50,964,5189,1745]);
// Polycyclic

G:=Group<a,b,c|a^12=b^5=1,c^4=a^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C12.F5 in TeX
Character table of C12.F5 in TeX

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