C₂×#20276973 #27539488

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#20276973

Splittings

Sylow

C₅₃ ⋊ C₂×#2249

Subgroups

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂²×D₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
hypercenter	C₂²×D₁₆	D₅₃
hypocenter	C₅₃	C₂×#2249
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

854 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 203 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) = 91
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 4 generators on 91 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)(56,57,58,59)(61,67)(66,79)(68,81)(69,78)(70,82)(75,90)(80,83)(84,89);(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)(60,62,71,85,65,74,77,88,64,73,86,91,76,87,63,72)(61,66,75,84,69,80,81,82,67,79,90,89,78,83,68,70);(57,59)(60,61)(62,70)(63,75)(64,67)(65,78)(66,72)(68,71)(69,76)(73,82)(74,89)(77,90)(79,88)(80,91)(81,86)(83,85)(84,87);(54,55)

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324964);