C₂×#20272830 #27299027

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20272830

Splittings

Sylow

C₅₃ ⋊ C₂×#1389

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133139
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#1389
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(56,58,59,70,60,69,74,85,62,72,76,68,61,71,75,78)(57,63,64,80,77,83,87,84,66,81,82,67,65,79,73,86);(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)(56,57,60,65,62,66,61,77)(58,67,69,84,72,80,71,86)(59,73,74,82,76,87,75,64)(63,78,79,70,81,85,83,68);(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, 84503);

C2×#20272830 #27299027