#27124809

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting	D₄×#41×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ #62811

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	D₄×#41×C₄₇	C₂
hypercenter	D₄×#41	D₄₇
hypocenter	C₄₇	#62811
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ D₄×#41

11534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 6 generators on 67 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)(52,55,56,63)(53,59,58,65)(54,62,61,67)(57,64,60,66);(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)(53,58)(57,60)(59,65)(64,66);(54,61)(57,60)(62,67)(64,66);(52,54)(53,57)(55,62)(56,61)(58,60)(59,64)(63,67)(65,66);(49,51);(48,49)(50,51)(52,53)(54,60)(55,59)(56,58)(57,61)(62,66)(63,65)(64,67)

Polycyclic presentation:

2^8,47|f^2=h,b^a=bg,c^a=ch,e^a=eh,d^c=dh,i^e=i^46

Conjugacy classes & character table

1238 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1078099);