C₂×#29503844 #44478948

order	30492
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂×C₁₁

Aliases

direct product

C₂×#29503844

Splittings

derived, Hall	C₇×C₁₁² ⋊ C₂²×C₉
Hall	C₇×C₁₁²⋊C₉ ⋊ C₂²
Hall	C₂×C₇⋊(C₁₁⋊D₁₁) ⋊ C₉
Hall	C₁₁²⋊C₉ ⋊ C₂×D₇
Sylow	C₁₁² ⋊ C₂×C₉×D₇
Sylow	C₇ ⋊ C₂×(C₁₁⋊D₁₁)⋊C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#19099626
commutator, hypocenter	C₇×C₁₁²	C₂²×C₉
Frattini	C₃	C₂×#19099626
Fitting, socle	C₂×C₃×C₇×C₁₁²	C₂×C₃
prime radical	C₂×C₃×C₇⋊(C₁₁⋊D₁₁)	C₃
cosocle	C₃×C₇×C₁₁²	C₂²×C₃
LCS	G ▷ C₇×C₁₁²
UCS	C₁ ◁ C₂×C₃

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1386

A minimal faithful permutation representation — 2 generators on 51 points:

(4,5)(6,7)(8,9)(10,14,18,12,17,13,15,11,16)(19,20,22)(21,33,35,25,26,42)(23,41,38,30,31,48)(24,29,39,34,28,40)(27,45,47,36,37,51)(32,46,50,43,44,49);(1,2)(3,4,6,8,9,7,5)(19,21,27,38,40,49,50,39,48,36,25)(20,23,32,45,26,34,24,33,37,43,30)(22,28,31,42,44,47,51,46,35,41,29)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30492, 171);

C2×#29503844 #44478948