C₂⁴×D₄×D₅₃ #28297968

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂⁴×D₄×D₅₃

Splittings

Fitting	C₂⁴×D₄×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂⁵×D₄

Subgroups

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂×C₅₃	C₂⁷
Frattini	C₂	C₂⁶×D₅₃
Fitting	C₂⁴×D₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
hypercenter	C₂⁴×D₄	D₅₃
hypocenter	C₅₃	C₂⁵×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×D₄

111198 conjugacy classes of subgroups; only the 34795 normal subgroups are shown.

normal subgroups only

This group has 111198 conjugacy classes of subgroups — too many to draw. Only its 34795 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) = 65
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 7 generators on 65 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)(63,65);(62,63)(64,65);(60,61);(58,59);(56,57);(54,55)

Conjugacy classes & character table

2240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 1083444);