C₂×D₄₄₂₅ #31799161

order	17700
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₈₈₅₀
direct product	C₂×D₄₄₂₅
unambiguous semidirect product	(C₂×C₃)⋊D₁₄₇₅
unambiguous semidirect product	(C₂×C₂₅)⋊D₁₇₇
unambiguous semidirect product	(C₂×C₅₉)⋊D₇₅

Splittings

Fitting	C₂×C₃×C₂₅×C₅₉ ⋊ C₂
derived, Hall	C₃×C₂₅×C₅₉ ⋊ C₂²
Hall	C₂₅×C₅₉ ⋊ C₂×D₃
Hall	C₃×C₅₉ ⋊ C₂×D₂₅
Hall	C₃×C₂₅ ⋊ C₂×D₅₉
Sylow	C₅₉ ⋊ C₂×D₇₅
Sylow	C₂₅ ⋊ C₂×D₁₇₇
Sylow	C₃ ⋊ C₂×D₁₄₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₄₂₅
commutator, cosocle, hypocenter	C₃×C₂₅×C₅₉	C₂²
Frattini	C₅	C₂×D₈₈₅
Fitting	C₂×C₃×C₂₅×C₅₉	C₂
socle	C₂×C₃×C₅×C₅₉	D₅
LCS	G ▷ C₃×C₂₅×C₅₉
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 8850

A minimal faithful permutation representation — 2 generators on 89 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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(4,5)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Conjugacy classes & character table

4428 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₇₀₀, C₂×D₁₃₂₇₅
maximal normal subgroup of	D₁₇₇₀₀
maximal quotient of	Dic₈₈₅₀, D₁₇₇₀₀, C₂×D₁₃₂₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17700, 22);

C2×D4425 #31799161