#27290664

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #10983

Subgroups

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator, prime radical	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#10983
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

1318 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 267 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) = 6784

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)(59,61)(62,65,67,64)(63,69,70,68)(66,72,74,73)(71,75,77,76);(55,57)(58,59)(60,61)(62,64,67,65)(63,68,70,69)(66,73,74,72)(71,76,77,75);(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)(56,57)(62,63,66,71,67,70,74,77)(64,69,72,76,65,68,73,75)

Polycyclic presentation:

2^8,53|a^2=g,b^2=h,c^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,i^a=i^52,c^b=cf,g^b=gh,g^c=gh

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, 76140);