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G = C3×C4⋊F5order 240 = 24·3·5

Direct product of C3 and C4⋊F5

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

Aliases: C3×C4⋊F5, C603C4, C123F5, C201C12, Dic53C12, C4⋊(C3×F5), C153(C4⋊C4), (C2×F5).C6, D5.(C3×Q8), C2.5(C6×F5), (C3×D5).4D4, (C4×D5).4C6, D5.1(C3×D4), (C6×F5).3C2, C6.18(C2×F5), (C3×D5).3Q8, C10.4(C2×C12), C30.18(C2×C4), (C3×Dic5)⋊7C4, D10.5(C2×C6), (D5×C12).11C2, (C6×D5).24C22, C5⋊(C3×C4⋊C4), SmallGroup(240,114)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C4⋊F5
C1C5C10D10C6×D5C6×F5 — C3×C4⋊F5
C5C10 — C3×C4⋊F5
C1C6C12

Generators and relations for C3×C4⋊F5
 G = < a,b,c,d | a3=b4=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

5C2
5C2
5C4
5C22
10C4
10C4
5C6
5C6
5C2×C4
5C2×C4
5C2×C4
5C12
5C2×C6
10C12
10C12
2F5
2F5
5C4⋊C4
5C2×C12
5C2×C12
5C2×C12
2C3×F5
2C3×F5
5C3×C4⋊C4

Smallest permutation representation of C3×C4⋊F5
On 60 points
Generators in S60
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)
(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)
(2 3 5 4)(6 8 7 10)(11 13 12 15)(16 18 17 20)(21 23 22 25)(26 28 27 30)(31 38 32 40)(33 37 35 36)(34 39)(41 48 42 50)(43 47 45 46)(44 49)(51 58 52 60)(53 57 55 56)(54 59)

G:=sub<Sym(60)| (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55), (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), (2,3,5,4)(6,8,7,10)(11,13,12,15)(16,18,17,20)(21,23,22,25)(26,28,27,30)(31,38,32,40)(33,37,35,36)(34,39)(41,48,42,50)(43,47,45,46)(44,49)(51,58,52,60)(53,57,55,56)(54,59)>;

G:=Group( (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55), (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), (2,3,5,4)(6,8,7,10)(11,13,12,15)(16,18,17,20)(21,23,22,25)(26,28,27,30)(31,38,32,40)(33,37,35,36)(34,39)(41,48,42,50)(43,47,45,46)(44,49)(51,58,52,60)(53,57,55,56)(54,59) );

G=PermutationGroup([[(1,24,14),(2,25,15),(3,21,11),(4,22,12),(5,23,13),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(1,39,9,34),(2,40,10,35),(3,36,6,31),(4,37,7,32),(5,38,8,33),(11,46,16,41),(12,47,17,42),(13,48,18,43),(14,49,19,44),(15,50,20,45),(21,56,26,51),(22,57,27,52),(23,58,28,53),(24,59,29,54),(25,60,30,55)], [(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)], [(2,3,5,4),(6,8,7,10),(11,13,12,15),(16,18,17,20),(21,23,22,25),(26,28,27,30),(31,38,32,40),(33,37,35,36),(34,39),(41,48,42,50),(43,47,45,46),(44,49),(51,58,52,60),(53,57,55,56),(54,59)]])

C3×C4⋊F5 is a maximal subgroup of
D60⋊C4  Dic6⋊F5  Dic5.Dic6  Dic5.4Dic6  C4⋊F53S3  Dic65F5  D603C4  C3×D4×F5  C3×Q8×F5

42 conjugacy classes

class 1 2A2B2C3A3B4A4B···4F 5 6A6B6C6D6E6F 10 12A12B12C···12L15A15B20A20B30A30B60A60B60C60D
order12223344···4566666610121212···1215152020303060606060
size115511210···10411555542210···104444444444

42 irreducible representations

dim11111111112222444444
type++++-++
imageC1C2C2C3C4C4C6C6C12C12D4Q8C3×D4C3×Q8F5C2×F5C3×F5C4⋊F5C6×F5C3×C4⋊F5
kernelC3×C4⋊F5D5×C12C6×F5C4⋊F5C3×Dic5C60C4×D5C2×F5Dic5C20C3×D5C3×D5D5D5C12C6C4C3C2C1
# reps11222224441122112224

Matrix representation of C3×C4⋊F5 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
2400
4500
1662
5061
,
1032
5261
2346
0526
,
6044
2202
5240
0322
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,4,1,5,4,5,6,0,0,0,6,6,0,0,2,1],[1,5,2,0,0,2,3,5,3,6,4,2,2,1,6,6],[6,2,5,0,0,2,2,3,4,0,4,2,4,2,0,2] >;

C3×C4⋊F5 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes F_5
% in TeX

G:=Group("C3xC4:F5");
// GroupNames label

G:=SmallGroup(240,114);
// by ID

G=gap.SmallGroup(240,114);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,151,3461,599]);
// Polycyclic

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

Export

Subgroup lattice of C3×C4⋊F5 in TeX

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