C₂×(C₁₃⋊C₉)⋊(C₇⋊D₇) #37524555

order	22932
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂×C₇

Aliases

direct product	C₂×(C₁₃⋊C₉)⋊(C₇⋊D₇)
unambiguous semidirect product	(C₂×C₁₃⋊C₉)⋊(C₇⋊D₇)

Splittings

derived, Hall	C₇²×C₁₃ ⋊ C₂²×C₉
Hall	C₇²×C₁₃⋊C₉ ⋊ C₂²
Hall	C₂×C₇⋊D₉₁ ⋊ C₉
Hall	C₁₃⋊C₉ ⋊ C₂×C₇⋊D₇
Sylow	C₇² ⋊ C₂×D₁₃⋊C₉
Sylow	C₁₃ ⋊ C₂×C₉×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₁₃⋊C₃)⋊(C₇⋊D₇)
commutator, hypocenter	C₇²×C₁₃	C₂²×C₉
Frattini	C₃	C₂×(C₁₃⋊C₃)⋊(C₇⋊D₇)
Fitting, socle	C₂×C₃×C₇²×C₁₃	C₂×C₃
prime radical	C₂×C₃×C₇⋊D₉₁	C₃
cosocle	C₃×C₇²×C₁₃	C₂²×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂×C₃

300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 11466

A minimal faithful permutation representation — 3 generators on 38 points:

(2,6,7,3,4,8)(5,12,11,9,10,13)(14,18,22,16,21,17,19,15,20)(24,25)(26,27)(28,29)(31,32)(33,34)(35,36);(1,2,5,4,8,11,10,12,13,7,6,9,3)(30,31,33,35,36,34,32)(37,38);(23,24,26,28,29,27,25)

Conjugacy classes & character table

1056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22932, 500);

C2×(C13⋊C9)⋊(C7⋊D7) #37524555