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## G = C5×C52⋊6C4order 500 = 22·53

### Direct product of C5 and C52⋊6C4

Aliases: C5×C526C4, C5311C4, C529C20, C528Dic5, C52(C5×Dic5), (C5×C10).9D5, C10.7(C5×D5), (C5×C10).8C10, C10.6(C5⋊D5), (C52×C10).2C2, C2.(C5×C5⋊D5), SmallGroup(500,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5×C52⋊6C4
 Chief series C1 — C5 — C52 — C5×C10 — C52×C10 — C5×C52⋊6C4
 Lower central C52 — C5×C52⋊6C4
 Upper central C1 — C10

Generators and relations for C5×C526C4
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 240 in 96 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C4, C5, C5, C5, C10, C10, C10, Dic5, C20, C52, C52, C52, C5×C10, C5×C10, C5×C10, C5×Dic5, C526C4, C53, C52×C10, C5×C526C4
Quotients: C1, C2, C4, C5, D5, C10, Dic5, C20, C5×D5, C5⋊D5, C5×Dic5, C526C4, C5×C5⋊D5, C5×C526C4

Smallest permutation representation of C5×C526C4
On 100 points
Generators in S100
(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)
(1 40 29 33 42)(2 36 30 34 43)(3 37 26 35 44)(4 38 27 31 45)(5 39 28 32 41)(6 22 13 98 20)(7 23 14 99 16)(8 24 15 100 17)(9 25 11 96 18)(10 21 12 97 19)(46 60 69 53 62)(47 56 70 54 63)(48 57 66 55 64)(49 58 67 51 65)(50 59 68 52 61)(71 87 78 94 85)(72 88 79 95 81)(73 89 80 91 82)(74 90 76 92 83)(75 86 77 93 84)
(1 30 44 38 32)(2 26 45 39 33)(3 27 41 40 34)(4 28 42 36 35)(5 29 43 37 31)(6 12 18 24 99)(7 13 19 25 100)(8 14 20 21 96)(9 15 16 22 97)(10 11 17 23 98)(46 70 64 58 52)(47 66 65 59 53)(48 67 61 60 54)(49 68 62 56 55)(50 69 63 57 51)(71 77 83 89 95)(72 78 84 90 91)(73 79 85 86 92)(74 80 81 87 93)(75 76 82 88 94)
(1 7 51 76)(2 8 52 77)(3 9 53 78)(4 10 54 79)(5 6 55 80)(11 60 85 35)(12 56 81 31)(13 57 82 32)(14 58 83 33)(15 59 84 34)(16 65 90 40)(17 61 86 36)(18 62 87 37)(19 63 88 38)(20 64 89 39)(21 70 95 45)(22 66 91 41)(23 67 92 42)(24 68 93 43)(25 69 94 44)(26 96 46 71)(27 97 47 72)(28 98 48 73)(29 99 49 74)(30 100 50 75)

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

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

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

140 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5BL 10A 10B 10C 10D 10E ··· 10BL 20A ··· 20H order 1 2 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 20 ··· 20 size 1 1 25 25 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 25 ··· 25

140 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C5 C10 C20 D5 Dic5 C5×D5 C5×Dic5 kernel C5×C52⋊6C4 C52×C10 C53 C52⋊6C4 C5×C10 C52 C5×C10 C52 C10 C5 # reps 1 1 2 4 4 8 12 12 48 48

Matrix representation of C5×C526C4 in GL5(𝔽41)

 10 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 37 0 0 0 0 0 37
,
 1 0 0 0 0 0 37 0 0 0 0 0 10 0 0 0 0 0 18 0 0 0 0 0 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 18
,
 9 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(41))| [10,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,37,0,0,0,0,0,37],[1,0,0,0,0,0,37,0,0,0,0,0,10,0,0,0,0,0,18,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,18],[9,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C5×C526C4 in GAP, Magma, Sage, TeX

C_5\times C_5^2\rtimes_6C_4
% in TeX

G:=Group("C5xC5^2:6C4");
// GroupNames label

G:=SmallGroup(500,38);
// by ID

G=gap.SmallGroup(500,38);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,1603,10004]);
// Polycyclic

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

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