D₅₃×#1335 #27293873

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₅₃×#1335

Splittings

Fitting	C₅₃×#1335 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1335

Subgroups

	subgroup	G/H
center	C₈	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1335	C₂
socle	C₂×C₅₃	C₂×D₄⋊C₈
prime radical	#135×D₅₃	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1335	D₅₃
hypocenter	C₅₃	C₂×#1335
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1335

414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 848

A minimal faithful permutation representation — 3 generators on 69 points:

(1,3)(2,7)(4,13)(5,10)(6,14)(8,16)(9,11)(12,15)(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);(1,2,4,9,6,10,16,12,7,3,11,13,5,14,15,8);(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

1232 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 79349);

D53×#1335 #27293873