#72209833

order	21504
isoclinism	isoclinic class of #72209833
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting

H(7,𝔽₂) ⋊ GL(3,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	#72209654
commutator, hypocenter	G	C₁
Fitting, cosocle	H(7,𝔽₂)	GL(3,𝔽₂)
LCS	G
UCS	C₁ ◁ C₂

2318 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 2318 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.

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

Full subgroup data →

Presentation

Permutation generators:

(2,6)(4,7)(8,13)(11,14);(2,7,3)(6,9,12)(8,14,10)(13,15,16);(2,8)(7,14)(9,15)(12,16);(3,10)(6,13)(7,14)(12,16);(4,11)(6,13)(9,15)(12,16);(1,2)(3,7)(4,9)(5,8)(6,12)(10,14)(11,15)(13,16);(1,3)(2,7)(4,6)(5,10)(8,14)(9,12)(11,13)(15,16);(1,4)(2,9)(3,6)(5,11)(7,12)(8,15)(10,13)(14,16);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	C₂×#72209833
maximal quotient of	C₂×#72209833, #72227702, #72227800, …
automorphism group of	C₂³×C₄
outer automorphism group of	C₂³×C₄

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((2,6)(4,7)(8,13)(11,14), (2,7,3)(6,9,12)(8,14,10)(13,15,16), (2,8)(7,14)(9,15)(12,16), (3,10)(6,13)(7,14)(12,16), (4,11)(6,13)(9,15)(12,16), (1,2)(3,7)(4,9)(5,8)(6,12)(10,14)(11,15)(13,16), (1,3)(2,7)(4,6)(5,10)(8,14)(9,12)(11,13)(15,16), (1,4)(2,9)(3,6)(5,11)(7,12)(8,15)(10,13)(14,16), (1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16));