C₂×D₆₉₇₅ #42124308

order	27900
automorphism	Aut(G) order 50220000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₃₉₅₀
direct product	C₂×D₆₉₇₅
unambiguous semidirect product	(C₂×C₉)⋊D₇₇₅
unambiguous semidirect product	(C₂×C₂₅)⋊D₂₇₉
unambiguous semidirect product	(C₂×C₃₁)⋊D₂₂₅

Splittings

Fitting	C₂×C₉×C₂₅×C₃₁ ⋊ C₂
derived, Hall	C₉×C₂₅×C₃₁ ⋊ C₂²
Hall	C₂₅×C₃₁ ⋊ C₂×D₉
Hall	C₉×C₃₁ ⋊ C₂×D₂₅
Hall	C₉×C₂₅ ⋊ C₂×D₃₁
Sylow	C₃₁ ⋊ C₂×D₂₂₅
Sylow	C₂₅ ⋊ C₂×D₂₇₉
Sylow	C₉ ⋊ C₂×D₇₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₉₇₅
commutator, cosocle, hypocenter	C₉×C₂₅×C₃₁	C₂²
Frattini	C₃×C₅	C₂×D₄₆₅
Fitting	C₂×C₉×C₂₅×C₃₁	C₂
socle	C₂×C₃×C₅×C₃₁	D₁₅
LCS	G ▷ C₉×C₂₅×C₃₁
UCS	C₁ ◁ C₂

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 13950

A minimal faithful permutation representation — 2 generators on 67 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);(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)(38,67)(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

6978 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₉₀₀
maximal normal subgroup of	D₂₇₉₀₀
maximal quotient of	Dic₁₃₉₅₀, D₂₇₉₀₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 55);

C2×D6975 #42124308