C₂×#20276133 #27412839

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20276133

Splittings

Sylow

C₅₃ ⋊ C₂×#2074

Subgroups

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#391	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂×#391	D₅₃
hypocenter	C₅₃	C₂×#2074
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#391

1062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 4 generators on 83 points:

(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)(57,59)(60,62,63,66)(61,65,64,67)(69,74)(70,76)(72,79)(73,80)(77,78)(81,83);(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)(61,64)(65,67)(68,70,71,77)(69,73,74,81)(72,76,79,78)(75,83,82,80);(56,57,58,59)(60,61)(62,65)(63,64)(66,67)(68,69)(70,73)(71,74)(72,75)(76,83)(77,81)(78,80)(79,82);(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, 198315);

C2×#20276133 #27412839