C₄×#16810891 #27390968

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×#16810891

Splittings

Fitting	C₄×AGL(1,ℤ/8)×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×#472

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133189
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×AGL(1,ℤ/8)×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#16810891	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×AGL(1,ℤ/8)	D₅₃
hypocenter	C₅₃	C₄×#472
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×AGL(1,ℤ/8)

946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 73 points:

(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)(58,60)(59,65)(61,68)(62,69)(63,70)(64,67)(66,73)(71,72);(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)(58,59,64,71,62,66,61,63)(60,65,67,72,69,73,68,70);(59,63)(60,67)(61,64)(66,71)(68,69)(70,72);(54,55,56,57)

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 176444);

C4×#16810891 #27390968