C₄×#16810814 #27390083

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₄×#16810814

Splittings

Fitting	C₄×#99×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×#458

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂⁴	C₂³×D₅₃
Fitting	C₄×#99×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#13133262	C₂²
cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₄×#99	D₅₃
hypocenter	C₅₃	C₄×#458
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₄×#99

1538 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)(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)(58,61)(59,64)(60,66)(62,68)(63,69)(65,71)(67,73)(70,72);(58,60)(59,63)(61,66)(62,67)(64,69)(65,70)(68,73)(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)(58,59,62,65)(60,63,67,70)(61,64,68,71)(66,72,73,69);(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, 175559);

C4×#16810814 #27390083