#27299149

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1375 ⋊ C₂
Sylow	C₅₃ ⋊ #12605

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1375	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×D₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1375	D₅₃
hypocenter	C₅₃	#12605
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #1375

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

normal subgroups only

This group has 510 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) = 73
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=e,d^2=h,f^2=g,g^2=h,b^a=bd,f^a=fh,c^b=cg,e^b=eh,f^c=fh,i^c=i^52

Conjugacy classes & character table

1208 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 84625);