#27079782

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄₇×#568 ⋊ C₂
Sylow	C₄₇ ⋊ #60076

Subgroups

	subgroup	G/H
center	C₄	C₂²×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#568	C₂
socle	C₂×C₄₇	C₂³×C₂²⋊C₄
prime radical	C₂×#15689959	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂×#568	D₄₇
hypocenter	C₄₇	#60076
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

4812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 5 generators on 79 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)(48,52,54,53)(49,57,58,59)(50,62,63,67)(51,65,64,66)(55,69,60,73)(56,71,70,72)(61,76,75,77)(68,78,74,79);(51,64)(56,70)(61,75)(65,66)(68,74)(71,72)(76,77)(78,79);(48,51)(49,56)(50,61)(52,65)(53,66)(54,64)(55,68)(57,71)(58,70)(59,72)(60,74)(62,76)(63,75)(67,77)(69,78)(73,79);(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)(49,55)(51,61)(52,62)(53,67)(54,63)(56,68)(57,69)(58,60)(59,73)(64,75)(65,76)(66,77)(70,74)(71,78)(72,79);(48,49,53,59,54,58,52,57)(50,60,62,73,63,55,67,69)(51,56,66,72,64,70,65,71)(61,74,76,79,75,68,77,78)

Polycyclic presentation:

2^8,47|a^2=g,e^2=h,g^2=h,b^a=bf,f^a=fh,g^b=gh,i^b=i^46,d^c=dh

Conjugacy classes & character table

1646 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033072);