#27083035

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×2₋¹⁺⁶ ⋊ C₂
Sylow	C₄₇ ⋊ #60400

Subgroups

	subgroup	G/H
center	C₂	C₂²×#13093993
commutator, cosocle	C₂³×C₄₇	C₂⁵
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₄₇×2₋¹⁺⁶	C₂
socle	C₂×C₄₇	C₂²×#120
hypercenter	2₋¹⁺⁶	D₄₇
hypocenter	C₄₇	#60400
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ 2₋¹⁺⁶

4732 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 807 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

1553 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1036325);