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G = C22×C5⋊F5  order 400 = 24·52

Direct product of C22 and C5⋊F5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C22×C5⋊F5, C102⋊7C4, (C2×C10)⋊3F5, C10⋊2(C2×F5), C5⋊2(C22×F5), C5⋊D5.5C23, C52⋊6(C22×C4), (C2×C5⋊D5)⋊8C4, C5⋊D5⋊5(C2×C4), (C5×C10)⋊5(C2×C4), (C22×C5⋊D5).6C2, (C2×C5⋊D5).27C22, SmallGroup(400,216)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C22×C5⋊F5
Chief series
C1 — C5 — C52 — C5⋊D5 — C5⋊F5 — C2×C5⋊F5 — C22×C5⋊F5
Lower central
C52 — C22×C5⋊F5
Upper central
C1 — C22

Generators and relations for C22×C5⋊F5
 G = < a,b,c,d,e | a2=b2=c5=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c3, ede-1=d3 >

Subgroups: 1272 in 216 conjugacy classes, 62 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C23, D5, C10, C22×C4, F5, D10, C2×C10, C52, C2×F5, C22×D5, C5⋊D5, C5⋊D5, C5×C10, C22×F5, C5⋊F5, C2×C5⋊D5, C102, C2×C5⋊F5, C22×C5⋊D5, C22×C5⋊F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, F5, C2×F5, C22×F5, C5⋊F5, C2×C5⋊F5, C22×C5⋊F5

Smallest permutation representation of C22×C5⋊F5
►On 100 points
Generators in S100
(1 81)(2 82)(3 83)(4 84)(5 85)(6 55)(7 51)(8 52)(9 53)(10 54)(11 60)(12 56)(13 57)(14 58)(15 59)(16 65)(17 61)(18 62)(19 63)(20 64)(21 70)(22 66)(23 67)(24 68)(25 69)(26 73)(27 74)(28 75)(29 71)(30 72)(31 76)(32 77)(33 78)(34 79)(35 80)(36 86)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)

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

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

40 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H5A···5F10A···10R
order122222224···45···510···10
size11112525252525···254···44···4

40 irreducible representations

dim1111144
type+++++
imageC1C2C2C4C4F5C2×F5
kernelC22×C5⋊F5C2×C5⋊F5C22×C5⋊D5C2×C5⋊D5C102C2×C10C10
# reps16162618

Matrix representation of C22×C5⋊F5 ►in GL12(ℤ)

-100000000000
0-10000000000
00-1000000000
000-100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
100000000000
010000000000
001000000000
000100000000
0000-10000000
00000-1000000
000000-100000
0000000-10000
000000001000
000000000100
000000000010
000000000001
,
010000000000
001000000000
000100000000
-1-1-1-100000000
000001000000
000000100000
000000010000
0000-1-1-1-10000
00000000000-1
00000000100-1
00000000010-1
00000000001-1
,
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
00000000000-1
00000000100-1
00000000010-1
00000000001-1
,
100000000000
000100000000
010000000000
-1-1-1-100000000
0000-10000000
0000000-10000
00000-1000000
000011110000
000000000010
000000001000
000000000001
000000000100

G:=sub<GL(12,Integers())| [-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,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1],[1,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0] >;
 

C22×C5⋊F5 in GAP, Magma, Sage, TeX 

C_2^2\times C_5\rtimes F_5
 
% in TeX
 
G:=Group("C2^2xC5:F5");
 
// GroupNames label
 
G:=SmallGroup(400,216);
 
// by ID
 
G=gap.SmallGroup(400,216);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,964,262,5765,1463]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^5=e^4=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,e*d*e^-1=d^3>;
 
// generators/relations
 

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