C₂×#20272657 #27293233

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20272657

Splittings

Sylow

C₅₃ ⋊ C₂×#1323

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#424	C₂
socle	C₂³×C₅₃	C₂²⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#424	D₅₃
hypocenter	C₅₃	C₂×#1323
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×#424

394 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 95
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 95 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)(56,58,59,63)(57,60,61,62)(64,66)(65,71)(67,80)(68,82)(69,77)(70,79)(72,88)(73,90)(74,76)(75,87)(78,84)(81,94)(83,85)(86,92)(89,93)(91,95);(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)(58,62,63,60)(59,61)(64,65,68,73,69,74,83,91,70,75,84,92,67,72,81,89)(66,76,78,93,77,87,94,90,79,88,82,95,80,71,85,86);(54,55)

Conjugacy classes & character table

1532 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78709);