#27081081

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	D₄×M₁₆×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ #60232

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁴×C₄
Fitting	D₄×M₁₆×C₄₇	C₂
prime radical	C₂×#15689670	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	D₄×M₁₆	D₄₇
hypocenter	C₄₇	#60232
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ D₄×M₁₆

2166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 71 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(57,62)(61,67)(63,69)(68,71);(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)(49,53)(52,54)(56,58)(57,61)(59,65)(60,64)(62,67)(63,68)(66,70)(69,71);(50,55)(52,54)(58,64)(61,67)(65,70)(68,71);(48,50)(49,52)(51,55)(53,54)(58,64)(61,67)(65,70)(68,71);(48,49)(50,54)(51,53)(52,55)(56,57,59,63,60,62,66,69)(58,61,65,68,64,67,70,71)

Polycyclic presentation:

2^8,47|a^2=g,g^2=h,b^a=bf,d^a=df,e^a=eh,c^b=cf,d^b=dh,d^c=dh,i^d=i^46

Conjugacy classes & character table

1226 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1034371);