C₂×#20274839 #27384265

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#20274839

Splittings

Sylow

C₅₃ ⋊ C₂×#1833

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	C₂×#1833
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 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)(56,59)(57,63,64,76)(58,67)(60,71)(61,72)(62,74,66,78)(65,70,77,85)(68,81)(69,82)(73,86)(75,84,79,80)(83,87);(56,58,60,68,61,69,73,83)(57,62,65,75,64,66,77,79)(59,67,71,81,72,82,86,87)(63,74,70,84,76,78,85,80);(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)(56,57)(58,66,69,62)(59,70,72,85)(60,65)(61,64)(63,71,76,86)(67,80)(68,79,83,75)(73,77)(74,81)(78,87)(82,84);(54,55)

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 169741);

C2×#20274839 #27384265