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## G = C22×C25⋊C4order 400 = 24·52

### Direct product of C22 and C25⋊C4

Aliases: C22×C25⋊C4, D503C4, D25.C23, D50.7C22, C50⋊(C2×C4), D25⋊(C2×C4), C25⋊(C22×C4), (C2×C50)⋊2C4, C5.(C22×F5), (C2×C10).4F5, C10.13(C2×F5), (C22×D25).3C2, SmallGroup(400,53)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C22×C25⋊C4
 Chief series C1 — C5 — C25 — D25 — C25⋊C4 — C2×C25⋊C4 — C22×C25⋊C4
 Lower central C25 — C22×C25⋊C4
 Upper central C1 — C22

Generators and relations for C22×C25⋊C4
G = < a,b,c,d | a2=b2=c25=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c18 >

Subgroups: 697 in 81 conjugacy classes, 37 normal (9 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C23, D5, C10, C22×C4, F5, D10, C2×C10, C25, C2×F5, C22×D5, D25, D25, C50, C22×F5, C25⋊C4, D50, C2×C50, C2×C25⋊C4, C22×D25, C22×C25⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, F5, C2×F5, C22×F5, C25⋊C4, C2×C25⋊C4, C22×C25⋊C4

Smallest permutation representation of C22×C25⋊C4
On 100 points
Generators in S100
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)(73 82)(74 83)(75 84)
(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 51)(2 58 25 69)(3 65 24 62)(4 72 23 55)(5 54 22 73)(6 61 21 66)(7 68 20 59)(8 75 19 52)(9 57 18 70)(10 64 17 63)(11 71 16 56)(12 53 15 74)(13 60 14 67)(26 94 27 76)(28 83 50 87)(29 90 49 80)(30 97 48 98)(31 79 47 91)(32 86 46 84)(33 93 45 77)(34 100 44 95)(35 82 43 88)(36 89 42 81)(37 96 41 99)(38 78 40 92)(39 85)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 5 10A 10B 10C 25A ··· 25E 50A ··· 50O order 1 2 2 2 2 2 2 2 4 ··· 4 5 10 10 10 25 ··· 25 50 ··· 50 size 1 1 1 1 25 25 25 25 25 ··· 25 4 4 4 4 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 F5 C2×F5 C25⋊C4 C2×C25⋊C4 kernel C22×C25⋊C4 C2×C25⋊C4 C22×D25 D50 C2×C50 C2×C10 C10 C22 C2 # reps 1 6 1 6 2 1 3 5 15

Matrix representation of C22×C25⋊C4 in GL5(𝔽101)

 100 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100
,
 1 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100
,
 1 0 0 0 0 0 35 8 31 63 0 38 73 46 69 0 32 70 4 78 0 23 55 93 27
,
 91 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 100 100 100 100

G:=sub<GL(5,GF(101))| [100,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100],[1,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100,0,0,0,0,0,100],[1,0,0,0,0,0,35,38,32,23,0,8,73,70,55,0,31,46,4,93,0,63,69,78,27],[91,0,0,0,0,0,1,0,0,100,0,0,0,1,100,0,0,0,0,100,0,0,1,0,100] >;

C22×C25⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_{25}\rtimes C_4
% in TeX

G:=Group("C2^2xC25:C4");
// GroupNames label

G:=SmallGroup(400,53);
// by ID

G=gap.SmallGroup(400,53);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,3364,1462,178,5765,1463]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^25=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^18>;
// generators/relations

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