D₅₃×#2205 #27538264

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

Aliases

direct product

D₅₃×#2205

Splittings

Fitting	C₅₃×#2205 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2205

Subgroups

	subgroup	G/H
center	C₂	B(4,𝔽₂)×D₅₃
commutator, cosocle	C₂×Q₈×C₅₃	C₂⁴
Frattini	C₂×Q₈	C₂³×D₅₃
Fitting	C₅₃×#2205	C₂
socle	C₂×C₅₃	C₂×B(4,𝔽₂)
prime radical	2₋¹⁺⁴×D₅₃	C₂²
hypercenter	#2205	D₅₃
hypocenter	C₅₃	C₂×#2205
LCS	G ▷ C₂×Q₈×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #2205

1136 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 848

A minimal faithful permutation representation — 4 generators on 69 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323740);