#69366

order	384
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₃₂ ⋊ C₂
Sylow	C₃ ⋊ #2258

Subgroups

	subgroup	G/H
center	C₃₂	C₂×D₃
commutator, socle	C₂×C₃	C₂²×C₁₆
Frattini	C₁₆	C₂²×D₃
Fitting	C₂×C₃×C₃₂	C₂
prime radical	#235	C₈
cosocle	C₃×C₁₆	C₂³
hypercenter	C₂×C₃₂	D₃
hypocenter	C₃	#2258
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3)(6,16)(11,26)(17,36)(18,37)(19,38)(27,46)(28,47)(29,48)(39,56)(40,57)(41,58)(49,61)(50,62)(51,63)(59,66)(64,67);(2,3)(4,6)(5,11)(7,17)(8,18)(9,19)(10,16)(12,27)(13,28)(14,29)(15,26)(20,39)(21,40)(22,36)(23,41)(24,37)(25,38)(30,49)(31,50)(32,46)(33,51)(34,47)(35,48)(42,59)(43,56)(44,57)(45,58)(52,64)(53,61)(54,62)(55,63)(60,66)(65,67);(4,5,7,12,8,13,20,30,9,14,21,31,23,33,42,52,10,15,22,32,24,34,43,53,25,35,44,54,45,55,60,65)(6,11,17,27,18,28,39,49,19,29,40,50,41,51,59,64,16,26,36,46,37,47,56,61,38,48,57,62,58,63,66,67)

Polycyclic presentation:

2^7,3|a^2=d,d^2=e,e^2=f,f^2=g,c^b=cg,h^b=h^2

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682999, #10683001, #10715749, …
maximal normal subgroup of	#10682999, #10683001, #10715749, …
maximal quotient of	#10715365, #10715366, #10715382, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1942);