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## G = C22×C52⋊C3order 300 = 22·3·52

### Direct product of C22 and C52⋊C3

Aliases: C22×C52⋊C3, C1021C3, (C5×C10)⋊2C6, C523(C2×C6), SmallGroup(300,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C22×C52⋊C3
 Chief series C1 — C52 — C52⋊C3 — C2×C52⋊C3 — C22×C52⋊C3
 Lower central C52 — C22×C52⋊C3
 Upper central C1 — C22

Generators and relations for C22×C52⋊C3
G = < a,b,c,d,e | a2=b2=c5=d5=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c3d3, ede-1=c-1d >

Smallest permutation representation of C22×C52⋊C3
On 60 points
Generators in S60
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)
(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 9 7 10 8)(16 17 18 19 20)(21 24 22 25 23)(31 32 33 34 35)(36 39 37 40 38)(46 47 48 49 50)(51 54 52 55 53)
(1 14 8)(2 15 10)(3 11 7)(4 12 9)(5 13 6)(16 26 22)(17 27 24)(18 28 21)(19 29 23)(20 30 25)(31 41 37)(32 42 39)(33 43 36)(34 44 38)(35 45 40)(46 56 52)(47 57 54)(48 58 51)(49 59 53)(50 60 55)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A ··· 5H 6A ··· 6F 10A ··· 10X order 1 2 2 2 3 3 5 ··· 5 6 ··· 6 10 ··· 10 size 1 1 1 1 25 25 3 ··· 3 25 ··· 25 3 ··· 3

44 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C2 C3 C6 C52⋊C3 C2×C52⋊C3 kernel C22×C52⋊C3 C2×C52⋊C3 C102 C5×C10 C22 C2 # reps 1 3 2 6 8 24

Matrix representation of C22×C52⋊C3 in GL4(𝔽31) generated by

 1 0 0 0 0 30 0 0 0 0 30 0 0 0 0 30
,
 30 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 2 0 0 0 0 8 0 0 0 0 2
,
 1 0 0 0 0 2 0 0 0 0 16 0 0 0 0 1
,
 25 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(31))| [1,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[30,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,2,0,0,0,0,8,0,0,0,0,2],[1,0,0,0,0,2,0,0,0,0,16,0,0,0,0,1],[25,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

C22×C52⋊C3 in GAP, Magma, Sage, TeX

C_2^2\times C_5^2\rtimes C_3
% in TeX

G:=Group("C2^2xC5^2:C3");
// GroupNames label

G:=SmallGroup(300,41);
// by ID

G=gap.SmallGroup(300,41);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,5,973,1439]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^5=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^3*d^3,e*d*e^-1=c^-1*d>;
// generators/relations

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