C₄×#11923520 #72259145

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₃

solvable

Aliases

direct product

C₄×#11923520

Splittings

Fitting

C₄×#585 ⋊ C₃⋊D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	A₄⋊S₄
commutator, hypocenter	Ω⁺(4,𝔽₃)	C₂²×C₄
Frattini, socle	C₂²	C₂²×A₄⋊S₄
Fitting	C₄×#585	C₃⋊D₃
prime radical	C₂×#11923520	C₂
cosocle	C₂×Ω⁺(4,𝔽₃)	C₂³
LCS	G ▷ Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₄²

3541 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 3541 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 52
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 52 points:

(5,52,41,29,40,48)(6,19,51,9,50,34)(7,47,10,49,36,13)(8,28,37,30,39,43)(11,27,35,14,18,44)(12,17,15,24,46,23)(16,42,45,21,26,38)(20,25,22,32,33,31);(7,17)(8,21)(9,23)(10,22)(12,25)(13,29)(14,31)(15,30)(18,38)(19,36)(20,37)(24,35)(26,48)(27,46)(28,47)(32,45)(33,42)(34,40)(39,41)(43,52)(44,50)(49,51);(1,2,3,4)(8,9,10)(13,14,15)(18,19,24)(21,22,23)(26,27,32)(29,30,31)(33,34,41)(35,36,38)(39,40,42)(43,44,51)(45,46,48)(49,50,52)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

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

gap> G := Group((5,52,41,29,40,48)(6,19,51,9,50,34)(7,47,10,49,36,13)(8,28,37,30,39,43)(11,27,35,14,18,44)(12,17,15,24,46,23)(16,42,45,21,26,38)(20,25,22,32,33,31), (7,17)(8,21)(9,23)(10,22)(12,25)(13,29)(14,31)(15,30)(18,38)(19,36)(20,37)(24,35)(26,48)(27,46)(28,47)(32,45)(33,42)(34,40)(39,41)(43,52)(44,50)(49,51), (1,2,3,4)(8,9,10)(13,14,15)(18,19,24)(21,22,23)(26,27,32)(29,30,31)(33,34,41)(35,36,38)(39,40,42)(43,44,51)(45,46,48)(49,50,52));