D₅₄₂₅ #24925746

order	10850
automorphism	Aut(G) order 19530000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₄₂₅
unambiguous semidirect product	C₇⋊D₇₇₅
unambiguous semidirect product	C₂₅⋊D₂₁₇
unambiguous semidirect product	C₃₁⋊D₁₇₅

Splittings

Fitting, derived, Hall	C₇×C₂₅×C₃₁ ⋊ C₂
Hall	C₂₅×C₃₁ ⋊ D₇
Hall	C₇×C₃₁ ⋊ D₂₅
Hall	C₇×C₂₅ ⋊ D₃₁
Sylow	C₃₁ ⋊ D₁₇₅
Sylow	C₂₅ ⋊ D₂₁₇
Sylow	C₇ ⋊ D₇₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₅×C₃₁	C₂
Frattini	C₅	D₁₀₈₅
socle	C₅×C₇×C₃₁	D₅
LCS	G ▷ C₇×C₂₅×C₃₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 5425

A minimal faithful permutation representation — 2 generators on 63 points:

(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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,32)(28,31)(29,30)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2,5,7,31,5|b^5=e^4,b^a=b^4e,c^a=c^6,d^a=d^30,e^a=e^4

Conjugacy classes & character table

2714 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁×D₁₇₅, D₇×D₇₇₅, D₂₅×D₂₁₇, …
maximal normal subgroup of	D₃₁×D₁₇₅, D₇×D₇₇₅, D₂₅×D₂₁₇, …
maximal quotient of	Dic₅₄₂₅, C₂×D₅₄₂₅, C₃×D₅₄₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10850, 9);

D5425 #24925746