C₂×#20283109 #27590185

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄³

Aliases

direct product

C₂×#20283109

Splittings

Fitting	C₂²×#127×C₅₃ ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2867

Subgroups

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂³×C₅₃	C₂⁵
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂²×#127×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#127	D₅₃
hypocenter	C₅₃	C₂×#2867
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

8094 conjugacy classes of subgroups; only the 1243 normal subgroups are shown.

normal subgroups only

This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 1243 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 5 generators on 67 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)(64,65,66,67);(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)(60,61,62,63);(56,57,58,59);(57,59)(61,63)(65,67);(54,55)

Conjugacy classes & character table

1544 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 375661);