C₂×D₄₁₈₅ #72213331

order	16740
automorphism	Aut(G) order 18079200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₈₃₇₀
direct product	C₂×D₄₁₈₅

Splittings

Fitting	C₂×C₅×C₂₇×C₃₁ ⋊ C₂
derived, Hall	C₅×C₂₇×C₃₁ ⋊ C₂²
Hall	C₂₇×C₃₁ ⋊ C₂×D₅
Hall	C₅×C₃₁ ⋊ C₂×D₂₇
Hall	C₅×C₂₇ ⋊ C₂×D₃₁
Sylow	C₃₁ ⋊ C₂×D₁₃₅
Sylow	C₂₇ ⋊ C₂×D₁₅₅
Sylow	C₅ ⋊ C₂×D₈₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₁₈₅
commutator, cosocle, hypocenter	C₅×C₂₇×C₃₁	C₂²
Frattini	C₉	C₂×D₄₆₅
Fitting	C₂×C₅×C₂₇×C₃₁	C₂
socle	C₂×C₃×C₅×C₃₁	D₉
LCS	G ▷ C₅×C₂₇×C₃₁
UCS	C₁ ◁ C₂

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 8370

A minimal faithful permutation representation — 2 generators on 65 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);(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)(31,34)(32,33)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Conjugacy classes & character table

4188 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₆₇₄₀, C₂×D₁₂₅₅₅
maximal normal subgroup of	D₁₆₇₄₀
maximal quotient of	Dic₈₃₇₀, D₁₆₇₄₀, C₂×D₁₂₅₅₅

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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65), (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)(31,34)(32,33)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51));

C2×D4185 #72213331