D₅₃×#1293 #27274054

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

D₅₃×#1293

Splittings

Fitting	C₅₃×#1293 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#1293

Subgroups

	subgroup	G/H
center	C₂²	#95×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1293	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×D₅₃	Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1293	D₅₃
hypocenter	C₅₃	C₂×#1293
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1293

906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 77 points:

(1,3)(2,5)(4,8)(6,7)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)(25,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27);(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24);(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)(70,71)(72,73)(74,75)(76,77)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 59530);

D53×#1293 #27274054