#27092519

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₄₇×#443 ⋊ C₂
Sylow	C₄₇ ⋊ #61105

Subgroups

	subgroup	G/H
center	C₄	D₄×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#443	C₂
socle	C₂×C₄₇	C₂×D₄²
hypercenter	C₂×#443	D₄₇
hypocenter	C₄₇	#61105
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

2620 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)(48,52,55,57)(49,50,59,62)(51,64,66,73)(53,67,54,68)(56,60,69,71)(58,61,74,78)(63,77,72,75)(65,76,70,79);(48,51)(49,58)(50,61)(52,64)(53,70)(54,65)(55,66)(56,72)(57,73)(59,74)(60,75)(62,78)(63,69)(67,79)(68,76)(71,77);(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)(49,57)(51,61)(52,59)(53,60)(54,71)(55,62)(56,68)(58,73)(63,79)(64,74)(65,77)(66,78)(67,69)(70,75)(72,76);(48,49)(50,52)(51,63)(53,69)(54,56)(55,59)(57,62)(58,65)(60,68)(61,76)(64,77)(66,72)(67,71)(70,74)(73,75)(78,79);(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)(49,56)(50,60)(53,54)(58,72)(59,69)(61,75)(62,71)(63,74)(65,70)(67,68)(76,79)(77,78)

Polycyclic presentation:

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

Conjugacy classes & character table

1346 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045809);