C₂×C₂₃×C₉₇⋊C₃ #27211461

order	13386
isoclinism	isoclinic class of C₉₇⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×C₂₃×C₉₇⋊C₃
unambiguous semidirect product	(C₂×C₉₇)⋊(C₃×C₂₃)
unambiguous semidirect product	(C₂×C₂₃×C₉₇)⋊C₃

Splittings

Fitting, socle, Hall	C₂×C₂₃×C₉₇ ⋊ C₃
derived, Sylow	C₉₇ ⋊ C₂×C₃×C₂₃
Hall	C₂₃×C₉₇ ⋊ C₂×C₃
Hall	C₂×C₉₇ ⋊ C₃×C₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₃	C₉₇⋊C₃
commutator, cosocle, hypocenter	C₉₇	C₂×C₃×C₂₃
Frattini	C₁	G
Fitting, socle	C₂×C₂₃×C₉₇	C₃
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂×C₂₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 4462

A minimal faithful permutation representation — 2 generators on 122 points:

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

Conjugacy classes & character table

1610 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×C₂₃×C₉₇⋊C₃, C₂₃×Dic₉₇⋊C₃, Dic₂₃×C₉₇⋊C₃, …
maximal normal subgroup of	C₄×C₂₃×C₉₇⋊C₃, C₂₃×Dic₉₇⋊C₃, Dic₂₃×C₉₇⋊C₃, …
maximal quotient of	C₄×C₂₃×C₉₇⋊C₃, C₂²×C₂₃×C₉₇⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13386, 2);

C2×C23×C97⋊C3 #27211461