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## G = C22×C5⋊F5order 400 = 24·52

### Direct product of C22 and C5⋊F5

Aliases: C22×C5⋊F5, C1027C4, (C2×C10)⋊3F5, C102(C2×F5), C52(C22×F5), C5⋊D5.5C23, C526(C22×C4), (C2×C5⋊D5)⋊8C4, C5⋊D55(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 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 5A ··· 5F 10A ··· 10R order 1 2 2 2 2 2 2 2 4 ··· 4 5 ··· 5 10 ··· 10 size 1 1 1 1 25 25 25 25 25 ··· 25 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 4 4 type + + + + + image C1 C2 C2 C4 C4 F5 C2×F5 kernel C22×C5⋊F5 C2×C5⋊F5 C22×C5⋊D5 C2×C5⋊D5 C102 C2×C10 C10 # reps 1 6 1 6 2 6 18

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

 -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 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 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 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
,
 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 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
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -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 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0

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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