C₂×D₃₂₂₅ #27166371

order	12900
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₄₅₀
direct product	C₂×D₃₂₂₅
unambiguous semidirect product	(C₂×C₃)⋊D₁₀₇₅
unambiguous semidirect product	(C₂×C₂₅)⋊D₁₂₉
unambiguous semidirect product	(C₂×C₄₃)⋊D₇₅

Splittings

Fitting	C₂×C₃×C₂₅×C₄₃ ⋊ C₂
derived, Hall	C₃×C₂₅×C₄₃ ⋊ C₂²
Hall	C₂₅×C₄₃ ⋊ C₂×D₃
Hall	C₃×C₄₃ ⋊ C₂×D₂₅
Hall	C₃×C₂₅ ⋊ C₂×D₄₃
Sylow	C₄₃ ⋊ C₂×D₇₅
Sylow	C₂₅ ⋊ C₂×D₁₂₉
Sylow	C₃ ⋊ C₂×D₁₀₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₂₂₅
commutator, cosocle, hypocenter	C₃×C₂₅×C₄₃	C₂²
Frattini	C₅	C₂×D₆₄₅
Fitting	C₂×C₃×C₂₅×C₄₃	C₂
socle	C₂×C₃×C₅×C₄₃	D₅
LCS	G ▷ C₃×C₂₅×C₄₃
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 6450

A minimal faithful permutation representation — 2 generators on 73 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);(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)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Conjugacy classes & character table

3228 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₂₉₀₀, C₂×D₁₆₁₂₅
maximal normal subgroup of	D₁₂₉₀₀
maximal quotient of	D₁₂₉₀₀, Dic₆₄₅₀, C₂×D₁₆₁₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 40);

C2×D3225 #27166371