C₇²×D₂₃ #14555909

order	2254
isoclinism	isoclinic class of D₂₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₇² · Schur multiplier (H₂) = C₇

Aliases

direct product	C₇²×D₂₃
unambiguous semidirect product	C₂₃⋊(C₂×C₇²)

Splittings

Fitting, socle, Hall	C₇²×C₂₃ ⋊ C₂
derived, Sylow	C₂₃ ⋊ C₂×C₇²

Subgroups

	subgroup	G/H
center, hypercenter	C₇²	D₂₃
commutator, cosocle, hypocenter	C₂₃	C₂×C₇²
Frattini	C₁	G
Fitting, socle	C₇²×C₂₃	C₂
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₇²

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 1127

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5,6,7)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(8,9,10,11,12,13,14)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₇×D₇×D₂₃, D₂₃×C₇⋊D₇, C₂×C₇²×D₂₃, …
maximal normal subgroup of	C₇×D₇×D₂₃, D₂₃×C₇⋊D₇, C₂×C₇²×D₂₃, …
maximal quotient of	C₇²×Dic₂₃, C₂×C₇²×D₂₃, C₇²×D₁₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2254, 6);

C72×D23 #14555909