C₂×#20272645 #27293174

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20272645

Splittings

Sylow

C₅₃ ⋊ C₂×#1317

Subgroups

	subgroup	G/H
center	C₂³	#13133132
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#381	C₂
socle	C₂³×C₅₃	#99
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
hypercenter	C₂×#381	D₅₃
hypocenter	C₅₃	C₂×#1317
LCS	G ▷ C₂³×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ C₂×#381

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

normal subgroups only

This group has 726 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) = 79
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 3 generators on 79 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)(64,66,68,74)(65,69,71,77)(67,73,76,79)(70,72,75,78);(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)(56,57,58,60,59,61,62,63)(64,65)(66,72)(67,75,76,70)(68,71)(69,73,77,79)(74,78);(54,55)

Conjugacy classes & character table

1508 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78650);