C₂²×#16810439 #27537047

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

Aliases

direct product

C₂²×#16810439

Splittings

Sylow

C₅₃ ⋊ C₂²×#362

Subgroups

	subgroup	G/H
center	C₂³	#13133138
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×Q₁₆×C₅₃	C₂
socle	C₂³×C₅₃	#102
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×Q₁₆	D₅₃
hypocenter	C₅₃	C₂²×#362
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

1134 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(58,60,63,72)(59,64,68,81)(61,74,76,87)(62,77,78,71)(65,82,84,85)(66,80,67,79)(69,73,86,83)(70,75,88,89);(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)(58,59,62,66,63,68,78,67)(60,69,71,82,72,86,77,85)(61,73,75,65,76,83,89,84)(64,74,79,70,81,87,80,88);(56,57);(54,55)

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 322523);

C22×#16810439 #27537047