C2^2xC25:C4 - GroupNames

G = C₂²×C₂₅⋊C₄ order 400 = 2⁴·5²

Direct product of C₂² and C₂₅⋊C₄

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C₂²×C₂₅⋊C₄, D₅₀⋊₃C₄, 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

G = C22×C25⋊C4 order 400 = 24·52

Direct product of C22 and C25⋊C4

G = C₂²×C₂₅⋊C₄ order 400 = 2⁴·5²

Direct product of C₂² and C₂₅⋊C₄