#27083995

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₄₇ ⋊ #60499

Subgroups

	subgroup	G/H
center	C₂²	C₂³×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄×2₋¹⁺⁴×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×D₄
prime radical	C₂×2₋¹⁺⁴×C₄₇	C₂²
hypercenter	C₄×2₋¹⁺⁴	D₄₇
hypocenter	C₄₇	#60499
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×2₋¹⁺⁴

1970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 87 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)(59,70)(64,76)(67,81)(71,83)(77,86)(78,84)(80,87)(82,85);(56,59)(57,64)(58,67)(60,70)(61,71)(62,76)(63,77)(65,78)(66,80)(68,81)(69,82)(72,83)(73,84)(74,85)(75,86)(79,87);(56,58,60,68)(57,63,62,75)(59,67,70,81)(61,74,72,69)(64,77,76,86)(65,79,73,66)(71,85,83,82)(78,87,84,80);(48,50,51,55)(49,52,53,54)(57,62)(63,75)(64,76)(65,73)(66,79)(77,86)(78,84)(80,87);(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)(48,49,51,53)(50,54,55,52)(56,57,60,62)(58,66,68,79)(59,64,70,76)(61,73,72,65)(63,74,75,69)(67,80,81,87)(71,84,83,78)(77,85,86,82)

Polycyclic presentation:

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

Conjugacy classes & character table

1625 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1037285);