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## G = C4×C52⋊C3order 300 = 22·3·52

### Direct product of C4 and C52⋊C3

Aliases: C4×C52⋊C3, C527C12, (C5×C20)⋊C3, (C5×C10).2C6, C2.(C2×C52⋊C3), (C2×C52⋊C3).3C2, SmallGroup(300,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C4×C52⋊C3
 Chief series C1 — C52 — C5×C10 — C2×C52⋊C3 — C4×C52⋊C3
 Lower central C52 — C4×C52⋊C3
 Upper central C1 — C4

Generators and relations for C4×C52⋊C3
G = < a,b,c,d | a4=b5=c5=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3c3, dcd-1=b-1c >

Smallest permutation representation of C4×C52⋊C3
On 60 points
Generators in S60
(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)
(1 2 3 4 5)(6 7 8 9 10)(11 14 12 15 13)(16 19 17 20 18)(31 32 33 34 35)(36 37 38 39 40)(41 44 42 45 43)(46 49 47 50 48)
(1 24 13)(2 25 15)(3 21 12)(4 22 14)(5 23 11)(6 26 17)(7 27 19)(8 28 16)(9 29 18)(10 30 20)(31 51 42)(32 52 44)(33 53 41)(34 54 43)(35 55 45)(36 56 47)(37 57 49)(38 58 46)(39 59 48)(40 60 50)

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

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

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

44 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A ··· 5H 6A 6B 10A ··· 10H 12A 12B 12C 12D 20A ··· 20P order 1 2 3 3 4 4 5 ··· 5 6 6 10 ··· 10 12 12 12 12 20 ··· 20 size 1 1 25 25 1 1 3 ··· 3 25 25 3 ··· 3 25 25 25 25 3 ··· 3

44 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 type + + image C1 C2 C3 C4 C6 C12 C52⋊C3 C2×C52⋊C3 C4×C52⋊C3 kernel C4×C52⋊C3 C2×C52⋊C3 C5×C20 C52⋊C3 C5×C10 C52 C4 C2 C1 # reps 1 1 2 2 2 4 8 8 16

Matrix representation of C4×C52⋊C3 in GL4(𝔽61) generated by

 50 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 58 0 0 0 6 34 0 0 0 0 58
,
 1 0 0 0 0 58 0 0 0 40 20 0 0 60 0 1
,
 13 0 0 0 0 1 57 0 0 0 60 1 0 0 60 0
G:=sub<GL(4,GF(61))| [50,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,58,6,0,0,0,34,0,0,0,0,58],[1,0,0,0,0,58,40,60,0,0,20,0,0,0,0,1],[13,0,0,0,0,1,0,0,0,57,60,60,0,0,1,0] >;

C4×C52⋊C3 in GAP, Magma, Sage, TeX

C_4\times C_5^2\rtimes C_3
% in TeX

G:=Group("C4xC5^2:C3");
// GroupNames label

G:=SmallGroup(300,15);
// by ID

G=gap.SmallGroup(300,15);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,5,30,1928,2859]);
// Polycyclic

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

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