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## G = He5⋊5C4order 500 = 22·53

### 1st semidirect product of He5 and C4 acting via C4/C2=C2

Aliases: He55C4, C522C20, C522Dic5, (C5×C10).C10, C526C4⋊C5, C10.2(C5×D5), (C5×C10).1D5, C2.(C52⋊C10), (C2×He5).1C2, C5.2(C5×Dic5), SmallGroup(500,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — He5⋊5C4
 Chief series C1 — C5 — C52 — C5×C10 — C2×He5 — He5⋊5C4
 Lower central C52 — He5⋊5C4
 Upper central C1 — C2

Generators and relations for He55C4
G = < a,b,c,d | a5=b5=c5=d4=1, cac-1=ab=ba, dad-1=ac3, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Smallest permutation representation of He55C4
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 48 55 36 64)(2 49 51 37 65)(3 50 52 38 61)(4 46 53 39 62)(5 47 54 40 63)(6 83 18 74 30)(7 84 19 75 26)(8 85 20 71 27)(9 81 16 72 28)(10 82 17 73 29)(11 66 21 94 77)(12 67 22 95 78)(13 68 23 91 79)(14 69 24 92 80)(15 70 25 93 76)(31 96 90 45 56)(32 97 86 41 57)(33 98 87 42 58)(34 99 88 43 59)(35 100 89 44 60)
(2 65 37 51 49)(3 38 50 61 52)(4 53 62 46 39)(5 47 54 40 63)(6 30 74 18 83)(7 75 84 26 19)(8 20 27 85 71)(9 81 16 72 28)(11 77 94 21 66)(12 95 67 78 22)(13 23 79 68 91)(14 69 24 92 80)(31 96 90 45 56)(33 58 42 87 98)(34 43 99 59 88)(35 89 60 100 44)
(1 41 29 15)(2 33 30 21)(3 59 26 67)(4 35 27 23)(5 45 28 14)(6 66 49 58)(7 12 50 43)(8 68 46 60)(9 80 47 90)(10 76 48 86)(11 51 42 83)(13 53 44 85)(16 24 40 31)(17 25 36 32)(18 77 37 87)(19 95 38 99)(20 79 39 89)(22 61 34 75)(52 88 84 78)(54 96 81 92)(55 97 82 93)(56 72 69 63)(57 73 70 64)(62 100 71 91)(65 98 74 94)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 10 10 2 2 2 2 type + + + - + - image C1 C2 C4 C5 C10 C20 C52⋊C10 He5⋊5C4 D5 Dic5 C5×D5 C5×Dic5 kernel He5⋊5C4 C2×He5 He5 C52⋊6C4 C5×C10 C52 C2 C1 C5×C10 C52 C10 C5 # reps 1 1 2 4 4 8 2 2 2 2 8 8

Matrix representation of He55C4 in GL12(𝔽41)

 6 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 35 6 0 0 0 0 0 0 0 0 0 0 35 40 0 0 0 0 0 0 0 0 0 0 0 0 40 35 0 0 0 0 0 0 0 0 0 0 6 35 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 40 0
,
 6 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 35 6 0 0 0 0 0 0 0 0 0 0 35 40 0 0 0 0 0 0 0 0 0 0 0 0 40 35 0 0 0 0 0 0 0 0 0 0 6 35 0 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 6
,
 9 0 0 0 0 0 0 0 0 0 0 0 13 32 0 0 0 0 0 0 0 0 0 0 0 0 13 39 0 0 0 0 0 0 0 0 0 0 2 28 0 0 0 0 0 0 0 0 0 0 0 0 13 39 0 0 0 0 0 0 0 0 0 0 2 28 0 0 0 0 0 0 0 0 0 0 0 0 13 39 0 0 0 0 0 0 0 0 0 0 2 28 0 0 0 0 0 0 0 0 0 0 0 0 13 39 0 0 0 0 0 0 0 0 0 0 2 28 0 0 0 0 0 0 0 0 0 0 0 0 13 39 0 0 0 0 0 0 0 0 0 0 2 28

G:=sub<GL(12,GF(41))| [6,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,35,35,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0],[6,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,6],[9,13,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28] >;

He55C4 in GAP, Magma, Sage, TeX

{\rm He}_5\rtimes_5C_4
% in TeX

G:=Group("He5:5C4");
// GroupNames label

G:=SmallGroup(500,8);
// by ID

G=gap.SmallGroup(500,8);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,1603,1208,10004]);
// Polycyclic

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

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