C₂×#983⋊A₄ #72225992

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

solvable

Aliases

direct product

C₂×#983⋊A₄

Splittings

Fitting

C₂²×H(5,𝔽₂) ⋊ C₃×D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×A₄×S₄
commutator, cosocle, hypocenter	#994	C₂³×C₃
Frattini	C₂	C₂²×A₄×S₄
Fitting	C₂²×H(5,𝔽₂)	C₃×D₃
hypercenter	C₂³	A₄×S₄
LCS	G ▷ #994
UCS	C₁ ◁ C₂² ◁ C₂³

2296 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 2296 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 50
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 50 points:

(1,2)(3,25,9)(4,47,38)(5,17,19)(7,11,22)(8,26,28)(12,50,41)(13,33,43)(14,46,34)(15,27,42)(16,44,36)(21,39,24)(23,30,32)(31,45,37)(35,48,49);(3,22,11,25)(4,16,12,35)(5,19,21,24)(6,9,10,7)(8,29,30,40)(13,38,34,41)(14,15,33,31)(17,20,39,18)(23,32,28,26)(27,50,45,47)(36,37,49,42)(43,44,46,48);(3,4)(5,13)(6,14)(7,15)(8,27)(9,31)(10,33)(11,12)(16,25)(17,36)(18,37)(19,41)(20,42)(21,34)(22,35)(23,43)(24,38)(26,44)(28,46)(29,47)(30,45)(32,48)(39,49)(40,50)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226211, #72226288, C₂×#72215280, …
maximal normal subgroup of	#72226211, #72226288, C₂×#72215280
maximal quotient of	#72226211, D₃×#983⋊A₄

GAP

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

gap> G := Group((1,2)(3,25,9)(4,47,38)(5,17,19)(7,11,22)(8,26,28)(12,50,41)(13,33,43)(14,46,34)(15,27,42)(16,44,36)(21,39,24)(23,30,32)(31,45,37)(35,48,49), (3,22,11,25)(4,16,12,35)(5,19,21,24)(6,9,10,7)(8,29,30,40)(13,38,34,41)(14,15,33,31)(17,20,39,18)(23,32,28,26)(27,50,45,47)(36,37,49,42)(43,44,46,48), (3,4)(5,13)(6,14)(7,15)(8,27)(9,31)(10,33)(11,12)(16,25)(17,36)(18,37)(19,41)(20,42)(21,34)(22,35)(23,43)(24,38)(26,44)(28,46)(29,47)(30,45)(32,48)(39,49)(40,50));