C₂×D₉₂₃ #16814828

order	3692
automorphism	Aut(G) order 1550640
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₈₄₆
direct product	C₂×D₉₂₃
unambiguous semidirect product	(C₂×C₁₃)⋊D₇₁
unambiguous semidirect product	(C₂×C₇₁)⋊D₁₃

Splittings

Fitting, socle	C₂×C₁₃×C₇₁ ⋊ C₂
derived, Hall	C₁₃×C₇₁ ⋊ C₂²
Sylow	C₇₁ ⋊ C₂×D₁₃
Sylow	C₁₃ ⋊ C₂×D₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₉₂₃
commutator, cosocle, hypocenter	C₁₃×C₇₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₃×C₇₁	C₂
LCS	G ▷ C₁₃×C₇₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 1846

A minimal faithful permutation representation — 2 generators on 86 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);(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(17,86)(18,85)(19,84)(20,83)(21,82)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(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

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

Relationships

maximal subgroup of	C₂×(C₁₃⋊C₃)⋊D₇₁, D₃×D₉₂₃, C₂×D₂₇₆₉, …
maximal normal subgroup of	C₂×(C₁₃⋊C₃)⋊D₇₁, C₂×C₃×D₉₂₃, C₂×C₁₃⋊(D₇₁⋊C₅), …
maximal quotient of	D₃×D₉₂₃, C₂×D₂₇₆₉, C₂×C₃×D₉₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3692, 9);

C2×D923 #16814828