Copied to
clipboard

## G = C3×C4⋊F5order 240 = 24·3·5

### Direct product of C3 and C4⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C3×C4⋊F5
 Chief series C1 — C5 — C10 — D10 — C6×D5 — C6×F5 — C3×C4⋊F5
 Lower central C5 — C10 — C3×C4⋊F5
 Upper central C1 — C6 — C12

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 >

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 2A 2B 2C 3A 3B 4A 4B ··· 4F 5 6A 6B 6C 6D 6E 6F 10 12A 12B 12C ··· 12L 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D order 1 2 2 2 3 3 4 4 ··· 4 5 6 6 6 6 6 6 10 12 12 12 ··· 12 15 15 20 20 30 30 60 60 60 60 size 1 1 5 5 1 1 2 10 ··· 10 4 1 1 5 5 5 5 4 2 2 10 ··· 10 4 4 4 4 4 4 4 4 4 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 type + + + + - + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 Q8 C3×D4 C3×Q8 F5 C2×F5 C3×F5 C4⋊F5 C6×F5 C3×C4⋊F5 kernel C3×C4⋊F5 D5×C12 C6×F5 C4⋊F5 C3×Dic5 C60 C4×D5 C2×F5 Dic5 C20 C3×D5 C3×D5 D5 D5 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 2 2 4 4 4 1 1 2 2 1 1 2 2 2 4

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

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 2 4 0 0 4 5 0 0 1 6 6 2 5 0 6 1
,
 1 0 3 2 5 2 6 1 2 3 4 6 0 5 2 6
,
 6 0 4 4 2 2 0 2 5 2 4 0 0 3 2 2
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

׿
×
𝔽