#27083730

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

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#2916 ⋊ C₂
Sylow	C₄₇ ⋊ #60472

Subgroups

	subgroup	G/H
center	C₂	C₂²×#13093999
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#2916	C₂
socle	C₂×C₄₇	C₂²×#121
hypercenter	#2916	D₄₇
hypocenter	C₄₇	#60472
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #2916

2578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 111 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)(52,72)(58,82)(64,92)(69,97)(73,74)(77,101)(80,104)(83,84)(87,102)(89,108)(93,94)(96,105)(98,99)(100,107)(106,111)(109,110);(48,52)(49,58)(50,64)(51,69)(53,73)(54,74)(55,72)(56,77)(57,80)(59,83)(60,84)(61,82)(62,87)(63,89)(65,93)(66,94)(67,92)(68,96)(70,98)(71,97)(75,99)(76,100)(78,102)(79,101)(81,104)(85,105)(86,106)(88,107)(90,109)(91,108)(95,110)(103,111);(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)(48,51)(49,57)(50,63)(52,69)(53,70)(54,75)(55,71)(56,76)(58,80)(59,68)(60,85)(61,81)(62,86)(64,89)(65,90)(66,95)(67,91)(72,97)(73,98)(74,99)(77,100)(78,103)(79,88)(82,104)(83,96)(84,105)(87,106)(92,108)(93,109)(94,110)(101,107)(102,111);(48,50,54,66,55,67,53,65)(49,56,60,78,61,79,59,62)(51,63,75,95,71,91,70,90)(52,64,74,94,72,92,73,93)(57,76,85,103,81,88,68,86)(58,77,84,102,82,101,83,87)(69,89,99,110,97,108,98,109)(80,100,105,111,104,107,96,106);(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)(48,49,55,61)(50,62,67,78)(51,68,71,85)(52,58,72,82)(53,60,54,59)(56,66,79,65)(57,75,81,70)(63,88,91,76)(64,87,92,102)(69,96,97,105)(73,84,74,83)(77,94,101,93)(80,99,104,98)(86,95,103,90)(89,107,108,100)(106,110,111,109)

Polycyclic presentation:

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

Conjugacy classes & character table

1070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1037020);