C₂×D₆₇₅ #72210163

order	2700
automorphism	Aut(G) order 486000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₃₅₀
direct product	C₂×D₆₇₅

Splittings

Fitting	C₂×C₂₅×C₂₇ ⋊ C₂
derived, Hall	C₂₅×C₂₇ ⋊ C₂²
Sylow	C₂₇ ⋊ C₂×D₂₅
Sylow	C₂₅ ⋊ C₂×D₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₇₅
commutator, cosocle, hypocenter	C₂₅×C₂₇	C₂²
Frattini	C₅×C₉	C₂×D₁₅
Fitting	C₂×C₂₅×C₂₇	C₂
socle	C₂×C₃×C₅	D₄₅
LCS	G ▷ C₂₅×C₂₇
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 1350

A minimal faithful permutation representation — 2 generators on 54 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);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₂₇₀₀, C₂×D₂₀₂₅, C₂×D₃₃₇₅, …
maximal normal subgroup of	D₂₇₀₀
maximal quotient of	D₂₇₀₀, C₂×D₂₀₂₅, Dic₁₃₅₀, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((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), (4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43));

C2×D675 #72210163