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## G = C22×He5order 500 = 22·53

### Direct product of C22 and He5

direct product, metabelian, nilpotent (class 2), monomial

Aliases: C22×He5, C102⋊C5, C5.1C102, (C5×C10)⋊2C10, C10.4(C5×C10), C522(C2×C10), (C2×C10).1C52, SmallGroup(500,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×He5
 Chief series C1 — C5 — C52 — He5 — C2×He5 — C22×He5
 Lower central C1 — C5 — C22×He5
 Upper central C1 — C2×C10 — C22×He5

Generators and relations for C22×He5
G = < a,b,c,d,e | a2=b2=c5=d5=e5=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 195 in 75 conjugacy classes, 45 normal (6 characteristic)
C1, C2, C22, C5, C5, C10, C10, C2×C10, C2×C10, C52, C5×C10, C102, He5, C2×He5, C22×He5
Quotients: C1, C2, C22, C5, C10, C2×C10, C52, C5×C10, C102, He5, C2×He5, C22×He5

Smallest permutation representation of C22×He5
On 100 points
Generators in S100
(1 86)(2 87)(3 88)(4 89)(5 90)(6 53)(7 54)(8 55)(9 51)(10 52)(11 56)(12 57)(13 58)(14 59)(15 60)(16 64)(17 65)(18 61)(19 62)(20 63)(21 67)(22 68)(23 69)(24 70)(25 66)(26 76)(27 77)(28 78)(29 79)(30 80)(31 71)(32 72)(33 73)(34 74)(35 75)(36 92)(37 93)(38 94)(39 95)(40 91)(41 83)(42 84)(43 85)(44 81)(45 82)(46 96)(47 97)(48 98)(49 99)(50 100)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 78)(7 79)(8 80)(9 76)(10 77)(11 81)(12 82)(13 83)(14 84)(15 85)(16 89)(17 90)(18 86)(19 87)(20 88)(21 92)(22 93)(23 94)(24 95)(25 91)(26 51)(27 52)(28 53)(29 54)(30 55)(31 46)(32 47)(33 48)(34 49)(35 50)(36 67)(37 68)(38 69)(39 70)(40 66)(41 58)(42 59)(43 60)(44 56)(45 57)(71 96)(72 97)(73 98)(74 99)(75 100)
(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 30 45 31 36)(2 26 41 32 37)(3 27 42 33 38)(4 28 43 34 39)(5 29 44 35 40)(6 15 99 24 16)(7 11 100 25 17)(8 12 96 21 18)(9 13 97 22 19)(10 14 98 23 20)(46 67 61 55 57)(47 68 62 51 58)(48 69 63 52 59)(49 70 64 53 60)(50 66 65 54 56)(71 92 86 80 82)(72 93 87 76 83)(73 94 88 77 84)(74 95 89 78 85)(75 91 90 79 81)
(1 39 2 35 33)(3 45 28 41 5)(4 26 40 38 30)(6 13 17 20 12)(7 10 96 15 97)(8 16 9 25 23)(11 14 21 99 22)(18 24 19 100 98)(27 31 43 32 29)(34 37 44 42 36)(46 60 47 54 52)(48 61 70 62 50)(49 68 56 59 67)(51 66 69 55 64)(53 58 65 63 57)(71 85 72 79 77)(73 86 95 87 75)(74 93 81 84 92)(76 91 94 80 89)(78 83 90 88 82)

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

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

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

116 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 5E ··· 5AB 10A ··· 10L 10M ··· 10CF order 1 2 2 2 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 size 1 1 1 1 1 1 1 1 5 ··· 5 1 ··· 1 5 ··· 5

116 irreducible representations

 dim 1 1 1 1 5 5 type + + image C1 C2 C5 C10 He5 C2×He5 kernel C22×He5 C2×He5 C102 C5×C10 C22 C2 # reps 1 3 24 72 4 12

Matrix representation of C22×He5 in GL6(𝔽11)

 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 1 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 3 0 0 0 0 0 0 5 0 1 0 0 0 0 0 0 9 0 0 0
,
 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0

G:=sub<GL(6,GF(11))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,5,0,0],[1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[5,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C22×He5 in GAP, Magma, Sage, TeX

C_2^2\times {\rm He}_5
% in TeX

G:=Group("C2^2xHe5");
// GroupNames label

G:=SmallGroup(500,35);
// by ID

G=gap.SmallGroup(500,35);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,613]);
// Polycyclic

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

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