#27293445

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #11498

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#11498
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 6784

A minimal faithful permutation representation — 3 generators on 101 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)(54,57,58,65)(55,60,61,67)(56,63,64,69)(59,66,62,68)(70,73,74,89,76,90,77,75)(71,79,91,99,82,98,80,81)(72,84,92,86,87,88,85,100)(78,93,83,94,95,97,96,101);(54,56,58,64)(55,59,61,62)(57,63,65,69)(60,66,67,68)(70,72,76,87)(71,78,82,95)(73,88,90,84)(74,92,77,85)(75,86,89,100)(79,97,98,93)(80,96,91,83)(81,94,99,101);(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)(56,62,64,59)(57,60)(58,61)(63,68,69,66)(65,67)(70,71,75,81,77,80,90,98,76,82,89,99,74,91,73,79)(72,83,86,97,85,78,84,94,87,96,100,93,92,95,88,101)

Polycyclic presentation:

2^8,53|a^2=e,b^2=f,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,i^a=i^52,c^b=ch,d^b=dh,e^b=efh

Conjugacy classes & character table

1520 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 78921);