C₂²⋊A₄×#3633 #72242926

order	6528
automorphism	?
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂²⋊A₄×#3633

Splittings

Fitting, socle, derived	C₂⁴×C₁₇ ⋊ C₃×C₈
Hall	C₂⁴×#3633 ⋊ C₃
Sylow	C₁₇ ⋊ C₈×C₂²⋊A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₂⁴×C₁₇	C₃×C₈
prime radical	D₁₇×C₂²⋊A₄	C₄
cosocle	C₂⁴×#595	C₂×C₃
LCS	G ▷ C₂⁴×C₁₇
UCS	C₁

752 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 204

A minimal faithful permutation representation — 3 generators on 25 points:

(2,3,4,6,5,7,8,10)(9,12,14,16,11,15,13,17)(19,20,21)(23,24,25);(1,2,6,11,8,13,17,15,3,7,12,16,14,4,9,10,5)(22,25)(23,24);(18,21)(19,20)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊A₄×AGL(1,𝔽₁₇), #1020×#3633, #72254555
maximal normal subgroup of	C₂²⋊A₄×AGL(1,𝔽₁₇), #1020×#3633, #72254555
maximal quotient of	#997×#3633

GAP

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

gap> G := Group((2,3,4,6,5,7,8,10)(9,12,14,16,11,15,13,17)(19,20,21)(23,24,25), (1,2,6,11,8,13,17,15,3,7,12,16,14,4,9,10,5)(22,25)(23,24), (18,21)(19,20));