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## G = He5⋊6C4order 500 = 22·53

### 2nd semidirect product of He5 and C4 acting via C4/C2=C2

Aliases: He56C4, C523Dic5, (C5×C10).3D5, C10.4(C5⋊D5), C2.(He5⋊C2), (C2×He5).2C2, C5.2(C526C4), SmallGroup(500,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — He5 — He5⋊6C4
 Chief series C1 — C5 — C52 — He5 — C2×He5 — He5⋊6C4
 Lower central He5 — He5⋊6C4
 Upper central C1 — C10

Generators and relations for He56C4
G = < a,b,c,d | a5=b5=c5=d4=1, cac-1=ab=ba, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of He56C4
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 32 41 30 38)(2 33 42 26 39)(3 34 43 27 40)(4 35 44 28 36)(5 31 45 29 37)(6 12 99 25 18)(7 13 100 21 19)(8 14 96 22 20)(9 15 97 23 16)(10 11 98 24 17)(46 62 55 68 60)(47 63 51 69 56)(48 64 52 70 57)(49 65 53 66 58)(50 61 54 67 59)(71 80 82 93 90)(72 76 83 94 86)(73 77 84 95 87)(74 78 85 91 88)(75 79 81 92 89)
(2 39 26 42 33)(3 27 34 40 43)(4 44 36 35 28)(5 31 45 29 37)(6 25 12 18 99)(7 100 19 13 21)(8 14 96 22 20)(10 17 24 98 11)(46 60 68 55 62)(47 69 63 56 51)(48 52 57 64 70)(49 65 53 66 58)(71 80 82 93 90)(73 87 95 84 77)(74 91 78 88 85)(75 81 89 79 92)
(1 97 50 72)(2 14 46 90)(3 7 47 92)(4 18 48 85)(5 24 49 77)(6 64 91 35)(8 60 93 39)(9 67 94 30)(10 53 95 45)(11 66 87 29)(12 52 88 44)(13 63 89 34)(15 59 86 38)(16 54 83 41)(17 65 84 31)(19 56 81 40)(20 68 82 26)(21 69 79 27)(22 55 80 42)(23 61 76 32)(25 57 78 36)(28 99 70 74)(33 96 62 71)(37 98 58 73)(43 100 51 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), (1,32,41,30,38)(2,33,42,26,39)(3,34,43,27,40)(4,35,44,28,36)(5,31,45,29,37)(6,12,99,25,18)(7,13,100,21,19)(8,14,96,22,20)(9,15,97,23,16)(10,11,98,24,17)(46,62,55,68,60)(47,63,51,69,56)(48,64,52,70,57)(49,65,53,66,58)(50,61,54,67,59)(71,80,82,93,90)(72,76,83,94,86)(73,77,84,95,87)(74,78,85,91,88)(75,79,81,92,89), (2,39,26,42,33)(3,27,34,40,43)(4,44,36,35,28)(5,31,45,29,37)(6,25,12,18,99)(7,100,19,13,21)(8,14,96,22,20)(10,17,24,98,11)(46,60,68,55,62)(47,69,63,56,51)(48,52,57,64,70)(49,65,53,66,58)(71,80,82,93,90)(73,87,95,84,77)(74,91,78,88,85)(75,81,89,79,92), (1,97,50,72)(2,14,46,90)(3,7,47,92)(4,18,48,85)(5,24,49,77)(6,64,91,35)(8,60,93,39)(9,67,94,30)(10,53,95,45)(11,66,87,29)(12,52,88,44)(13,63,89,34)(15,59,86,38)(16,54,83,41)(17,65,84,31)(19,56,81,40)(20,68,82,26)(21,69,79,27)(22,55,80,42)(23,61,76,32)(25,57,78,36)(28,99,70,74)(33,96,62,71)(37,98,58,73)(43,100,51,75)>;

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

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

44 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5P 10A 10B 10C 10D 10E ··· 10P 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 10 ··· 10 1 1 1 1 10 ··· 10 25 ··· 25

44 irreducible representations

 dim 1 1 1 2 2 5 5 type + + + - image C1 C2 C4 D5 Dic5 He5⋊C2 He5⋊6C4 kernel He5⋊6C4 C2×He5 He5 C5×C10 C52 C2 C1 # reps 1 1 2 12 12 8 8

Matrix representation of He56C4 in GL7(𝔽41)

 0 40 0 0 0 0 0 1 6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 37 0 0 0 0 0 0 0 10 0 0 16 0 0 0 0
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 18
,
 35 6 0 0 0 0 0 35 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 37 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 16
,
 13 9 0 0 0 0 0 13 28 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0

G:=sub<GL(7,GF(41))| [0,1,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,10,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18],[35,35,0,0,0,0,0,6,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,16],[13,13,0,0,0,0,0,9,28,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0] >;

He56C4 in GAP, Magma, Sage, TeX

{\rm He}_5\rtimes_6C_4
% in TeX

G:=Group("He5:6C4");
// GroupNames label

G:=SmallGroup(500,11);
// by ID

G=gap.SmallGroup(500,11);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1603,613]);
// 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^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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