H(3,𝔽₄)×Dic₅₃ #27546798

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

direct product

H(3,𝔽₄)×Dic₅₃

Splittings

Sylow

C₅₃ ⋊ C₄×H(3,𝔽₄)

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting, prime radical	C₂×C₅₃×H(3,𝔽₄)	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×H(3,𝔽₄)	D₅₃
hypocenter	C₅₃	C₄×H(3,𝔽₄)
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×H(3,𝔽₄)

2664 conjugacy classes of subgroups; only the 1093 normal subgroups are shown.

normal subgroups only

This group has 2664 conjugacy classes of subgroups — too many to draw. Only its 1093 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) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 5 generators on 73 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55,56,57);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(59,63)(60,65)(67,71)(70,73);(60,65)(62,64)(67,71)(68,72);(58,60,61,65)(59,62,63,64)(66,68)(67,70)(69,72)(71,73);(58,59,61,63)(60,64,65,62)(66,67)(68,70)(69,71)(72,73)

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 332274);