#27299604

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅₃×#356 ⋊ C₂
Sylow	C₅₃ ⋊ #12701

Subgroups

	subgroup	G/H
center	C₂²	#99×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×Q₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#356	C₂
socle	C₂²×C₅₃	C₂×#99
prime radical	C₂×#16811174	C₂
cosocle	C₂²×Q₈×C₅₃	C₂³
hypercenter	C₂×#356	D₅₃
hypocenter	C₅₃	#12701
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ C₂×#356

1070 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 111 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:

(1,2)(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)(55,57)(58,60,63,62)(59,65,68,67)(61,69,70,71)(64,72,66,73);(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)(61,70)(65,72)(66,68)(67,73)(69,71);(54,55)(56,57)(58,59,62,67,63,68,60,65)(61,66,69,73,70,64,71,72)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fgh,i^b=i^52,i^c=i^52,f^d=fh

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85080);