C₂×(C₁₁⋊C₅)⋊D₁₁₃ #38733251

order	24860
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×(C₁₁⋊C₅)⋊D₁₁₃
unambiguous semidirect product	(C₂×C₁₁⋊C₅)⋊D₁₁₃
unambiguous semidirect product	(C₂×C₁₁₃)⋊AGL(1,𝔽₁₁)
unambiguous semidirect product	(C₂×D₁₂₄₃)⋊C₅

Splittings

Fitting, socle	C₂×C₁₁×C₁₁₃ ⋊ C₂×C₅
derived, Hall	C₁₁×C₁₁₃ ⋊ C₂²×C₅
Hall	C₁₁⋊C₅×C₁₁₃ ⋊ C₂²
Hall	C₂×D₁₂₄₃ ⋊ C₅
Sylow	C₁₁₃ ⋊ C₂×AGL(1,𝔽₁₁)
Hall	C₁₁⋊C₅ ⋊ C₂×D₁₁₃
Sylow	C₁₁ ⋊ C₂×C₅×D₁₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	(C₁₁⋊C₅)⋊D₁₁₃
commutator, cosocle, hypocenter	C₁₁×C₁₁₃	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₁₁×C₁₁₃	C₂×C₅
LCS	G ▷ C₁₁×C₁₁₃
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 126
transitive degree	μ_t(G) = 2486

A minimal faithful permutation representation — 2 generators on 126 points:

(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)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(115,119,120,118,123,116,117,124,121,122);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(114,115,118,123,117,120,124,119,122,121,116)(125,126)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24860, 18);

C2×(C11⋊C5)⋊D113 #38733251