C₂³×C₃⋊(C₇⋊D₇) #72245658

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₇

Aliases

direct product

C₂³×C₃⋊(C₇⋊D₇)

Splittings

Fitting, socle	C₂³×C₃×C₇² ⋊ C₂
derived, Hall	C₃×C₇² ⋊ C₂⁴
Sylow	C₇² ⋊ C₂³×D₃
Sylow	C₃ ⋊ C₂³×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	C₃⋊(C₇⋊D₇)
commutator, cosocle, hypocenter	C₃×C₇²	C₂⁴
Frattini	C₁	G
Fitting, socle	C₂³×C₃×C₇²	C₂
LCS	G ▷ C₃×C₇²
UCS	C₁ ◁ C₂³

1340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 1176

A minimal faithful permutation representation — 4 generators on 23 points:

(5,6)(7,8,9)(10,11,13,15,16,14,12);(1,2)(3,4)(5,6)(8,9)(11,12)(13,14)(15,16)(18,19)(20,21)(22,23);(3,4)(17,18,20,22,23,21,19);(1,2)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₄×C₃⋊(C₇⋊D₇), C₂³×D₃×C₇⋊D₇, C₂×#72245569, …
maximal normal subgroup of	C₂×D₄×C₃⋊(C₇⋊D₇), C₂³×D₃×C₇⋊D₇, C₂×#72245569, …
maximal quotient of	C₂×D₄×C₃⋊(C₇⋊D₇)

GAP

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

gap> G := Group((5,6)(7,8,9)(10,11,13,15,16,14,12), (1,2)(3,4)(5,6)(8,9)(11,12)(13,14)(15,16)(18,19)(20,21)(22,23), (3,4)(17,18,20,22,23,21,19), (1,2));

C23×C3⋊(C7⋊D7) #72245658