#27293409

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11487

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₈×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#11487
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

430 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 155 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) = 77
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|a^2=e,c^2=fg,e^2=g,g^2=h,b^a=bd,d^a=df,i^a=i^52,e^b=ef

Conjugacy classes & character table

2168 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78885);