C₇²×Dic₂₃ #17985196

order	4508
isoclinism	isoclinic class of D₂₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₇² · Schur multiplier (H₂) = C₇

Aliases

direct product	C₇²×Dic₂₃
unambiguous semidirect product	C₂₃⋊(C₄×C₇²)

Splittings

derived, Sylow	C₂₃ ⋊ C₄×C₇²
Hall	C₇²×C₂₃ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇²	D₂₃
commutator, hypocenter	C₂₃	C₄×C₇²
Frattini	C₂	C₇²×D₂₃
Fitting, socle, prime radical	C₂×C₇²×C₂₃	C₂
cosocle	C₂×C₂₃	C₂×C₇²
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₂×C₇²

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 4508

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5,6,7)(15,16)(17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(8,9,10,11,12,13,14)(15,17,16,18)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Conjugacy classes & character table

1274 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×ΓL(1,𝔽₈)×Dic₂₃, Dic₂₃×#3861, Dic₂₃×#3862, …
maximal normal subgroup of	C₇×ΓL(1,𝔽₈)×Dic₂₃, Dic₂₃×#3861, Dic₂₃×#3862, …
maximal quotient of	C₃×C₇²×Dic₂₃, C₇²×Dic₆₉, C₇²×Dic₁₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4508, 11);

C72×Dic23 #17985196