C₄²×C₈×D₅₃ #27381410

order	13568
isoclinism	isoclinic class of D₅₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₈ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

direct product

C₄²×C₈×D₅₃

Splittings

Fitting	C₄²×C₈×C₅₃ ⋊ C₂
derived, Sylow	C₅₃ ⋊ C₂×C₄²×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄²×C₈	D₅₃
commutator, hypocenter	C₅₃	C₂×C₄²×C₈
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄²×C₈

2046 conjugacy classes of subgroups; only the 1227 normal subgroups are shown.

normal subgroups only

This group has 2046 conjugacy classes of subgroups — too many to draw. Only its 1227 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) = 69
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 69 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);(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)(62,63,64,66,65,67,68,69);(58,59,60,61);(54,55,56,57)

Conjugacy classes & character table

3584 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 166886);