C₂×#20274827 #27383957

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

Aliases

direct product

C₂×#20274827

Splittings

Sylow

C₅₃ ⋊ C₂×#1830

Subgroups

	subgroup	G/H
center	C₂²	#16810484
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#101×C₅₃	C₂
socle	C₂²×C₅₃	#381
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#101	D₅₃
hypocenter	C₅₃	C₂×#1830
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#101

1218 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:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 169433);

C2×#20274827 #27383957