C₄×#10596399 #72259141

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²×C₃

solvable

Aliases

direct product

C₄×#10596399

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	A₄⋊S₄
commutator, hypocenter	Ω⁺(4,𝔽₃)	C₂×C₄
Frattini, socle	C₂²	C₂×A₄⋊S₄
Fitting	C₄×H(5,𝔽₂)	C₃⋊D₃
prime radical, cosocle	C₂×Ω⁺(4,𝔽₃)	C₂²
LCS	G ▷ Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₂×C₄

696 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 43 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:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#11923520
maximal normal subgroup of	C₄×#11923520

GAP

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

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