C₂×D₆₆₇₅ #40998887

order	26700
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₃₃₅₀
direct product	C₂×D₆₆₇₅
unambiguous semidirect product	(C₂×C₃)⋊D₂₂₂₅
unambiguous semidirect product	(C₂×C₂₅)⋊D₂₆₇
unambiguous semidirect product	(C₂×C₈₉)⋊D₇₅

Splittings

Fitting	C₂×C₃×C₂₅×C₈₉ ⋊ C₂
derived, Hall	C₃×C₂₅×C₈₉ ⋊ C₂²
Hall	C₂₅×C₈₉ ⋊ C₂×D₃
Hall	C₃×C₈₉ ⋊ C₂×D₂₅
Sylow	C₈₉ ⋊ C₂×D₇₅
Hall	C₃×C₂₅ ⋊ C₂×D₈₉
Sylow	C₂₅ ⋊ C₂×D₂₆₇
Sylow	C₃ ⋊ C₂×D₂₂₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₆₇₅
commutator, cosocle, hypocenter	C₃×C₂₅×C₈₉	C₂²
Frattini	C₅	C₂×D₁₃₃₅
Fitting	C₂×C₃×C₂₅×C₈₉	C₂
socle	C₂×C₃×C₅×C₈₉	D₅
LCS	G ▷ C₃×C₂₅×C₈₉
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 13350

A minimal faithful permutation representation — 2 generators on 119 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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(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)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Conjugacy classes & character table

6678 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₆₇₀₀
maximal normal subgroup of	D₂₆₇₀₀
maximal quotient of	Dic₁₃₃₅₀, D₂₆₇₀₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26700, 30);

C2×D6675 #40998887