C₄×#16810897 #27390981

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

Aliases

direct product

C₄×#16810897

Splittings

Fitting	C₄×#101×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₄×#473

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133190
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#101×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#11719945	C₂³
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#101	D₅₃
hypocenter	C₅₃	C₄×#473
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#101

882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 89 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)(59,66)(60,71)(62,76)(63,78)(64,77)(65,79)(67,75)(68,83)(69,82)(70,84)(72,86)(73,88)(74,87)(80,89)(81,85);(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)(58,60,63,73,64,74,62,72)(59,65,68,80,69,81,67,70)(61,71,78,88,77,87,76,86)(66,79,83,89,82,85,75,84);(58,59,64,69)(60,70,74,80)(61,75,77,83)(62,68,63,67)(65,73,81,72)(66,78,82,76)(71,85,87,79)(84,88,89,86);(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, 176457);

C4×#16810897 #27390981