C₂×D₄₂₇₅ #30685403

order	17100
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₈₅₅₀
direct product	C₂×D₄₂₇₅
unambiguous semidirect product	(C₂×C₉)⋊D₄₇₅
unambiguous semidirect product	(C₂×C₁₉)⋊D₂₂₅
unambiguous semidirect product	(C₂×C₂₅)⋊D₁₇₁

Splittings

Fitting	C₂×C₉×C₁₉×C₂₅ ⋊ C₂
derived, Hall	C₉×C₁₉×C₂₅ ⋊ C₂²
Hall	C₁₉×C₂₅ ⋊ C₂×D₉
Hall	C₉×C₂₅ ⋊ C₂×D₁₉
Hall	C₉×C₁₉ ⋊ C₂×D₂₅
Sylow	C₂₅ ⋊ C₂×D₁₇₁
Sylow	C₁₉ ⋊ C₂×D₂₂₅
Sylow	C₉ ⋊ C₂×D₄₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₂₇₅
commutator, cosocle, hypocenter	C₉×C₁₉×C₂₅	C₂²
Frattini	C₃×C₅	C₂×D₂₈₅
Fitting	C₂×C₉×C₁₉×C₂₅	C₂
socle	C₂×C₃×C₅×C₁₉	D₁₅
LCS	G ▷ C₉×C₁₉×C₂₅
UCS	C₁ ◁ C₂

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 8550

A minimal faithful permutation representation — 2 generators on 55 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);(4,11)(5,10)(6,9)(7,8)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Conjugacy classes & character table

4278 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₁₀₀, C₂×D₁₂₈₂₅
maximal normal subgroup of	D₁₇₁₀₀
maximal quotient of	Dic₈₅₅₀, D₁₇₁₀₀, C₂×D₁₂₈₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17100, 40);

C2×D4275 #30685403