C₂×D₃₀₇₅ #27134180

order	12300
automorphism	Aut(G) order 9840000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₁₅₀
direct product	C₂×D₃₀₇₅
unambiguous semidirect product	(C₂×C₃)⋊D₁₀₂₅
unambiguous semidirect product	(C₂×C₂₅)⋊D₁₂₃
unambiguous semidirect product	(C₂×C₄₁)⋊D₇₅

Splittings

Fitting	C₂×C₃×C₂₅×C₄₁ ⋊ C₂
derived, Hall	C₃×C₂₅×C₄₁ ⋊ C₂²
Hall	C₂₅×C₄₁ ⋊ C₂×D₃
Hall	C₃×C₄₁ ⋊ C₂×D₂₅
Hall	C₃×C₂₅ ⋊ C₂×D₄₁
Sylow	C₄₁ ⋊ C₂×D₇₅
Sylow	C₂₅ ⋊ C₂×D₁₂₃
Sylow	C₃ ⋊ C₂×D₁₀₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₀₇₅
commutator, cosocle, hypocenter	C₃×C₂₅×C₄₁	C₂²
Frattini	C₅	C₂×D₆₁₅
Fitting	C₂×C₃×C₂₅×C₄₁	C₂
socle	C₂×C₃×C₅×C₄₁	D₅
LCS	G ▷ C₃×C₂₅×C₄₁
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 6150

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

Conjugacy classes & character table

3078 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₉₂₂₅, D₃×D₃₀₇₅, C₂×C₃⋊D₃₀₇₅, …
maximal normal subgroup of	C₂×C₃×D₃₀₇₅, D₁₂₃₀₀
maximal quotient of	C₂×D₉₂₂₅, D₃×D₃₀₇₅, C₂×C₃⋊D₃₀₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12300, 42);

C2×D3075 #27134180