C₂×#20272699 #27294232

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

Aliases

direct product

C₂×#20272699

Splittings

Fitting	C₂×C₅₃×H(3,ℤ/4) ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1339

Subgroups

	subgroup	G/H
center	C₂²	#16810356
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×H(3,ℤ/4)	C₂
socle	C₂²×C₅₃	#327
prime radical	C₂×#16810876	C₂
cosocle	C₂×M₁₆×C₅₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₅₃
hypocenter	C₅₃	C₂×#1339
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

496 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 71
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 71 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)(57,61)(58,67)(59,68)(62,64)(63,70)(65,66)(69,71);(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)(56,57)(58,66,69,61)(59,63)(60,65)(62,67,70,71)(64,68);(54,55)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79708);