#27079654

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #60049

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₇
commutator, socle, cosocle	C₂³×C₄₇	C₂⁵
Fitting	C₄₇×#2296	C₂
prime radical	C₂⁴×C₄₇	C₂⁴
hypercenter	#2296	D₄₇
hypocenter	C₄₇	#60049
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #2296

1634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 6016

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)(49,53)(52,54)(57,62)(61,63)(64,67,68,66)(65,70,71,69);(49,53)(52,54)(64,66,68,67)(65,70,71,69);(49,53)(50,55)(56,59,60,58)(57,61,62,63)(64,66,68,67)(65,69,71,70);(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)(48,50)(49,52)(51,55)(53,54)(56,58,60,59)(57,61,62,63)(64,65)(66,69)(67,70)(68,71);(48,49,51,53)(50,54,55,52)(56,57,60,62)(58,63,59,61)

Polycyclic presentation:

2^8,47|a^2=f,b^2=fg,c^2=fgh,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,i^b=i^46

Conjugacy classes & character table

1352 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1032944);