C₄×#16810491 #27305328

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₄×#16810491

Splittings

Sylow

C₅₃ ⋊ C₄×#383

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₄×#383
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

1566 conjugacy classes of subgroups; only the 635 normal subgroups are shown.

normal subgroups only

This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 635 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) = 77
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 77 points:

(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)(63,67)(66,69)(70,72,73,77)(71,75,74,76);(62,64,65,68)(63,66,67,69)(71,74)(75,76);(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)(58,59,60,61)(62,63)(64,66)(65,67)(68,69)(70,71)(72,76)(73,74)(75,77);(54,55,56,57)

Conjugacy classes & character table

3440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90804);