C₂×#11714183 #72343193

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#11714183

Splittings

Fitting

C₂×#3026 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini	C₂³	C₂³×S₄
Fitting	C₂×#3026	D₃
hypercenter	C₂³	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂³

3300 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 3300 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236902, C₂×#72236922, C₂×#72236841
maximal normal subgroup of	C₂×#72236902, C₂×#72236922, C₂×#72236841

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,15,6,30,8,17,19,29)(4,25,11,34,9,26,22,33)(5,21,14,18,16,7,20,31)(10,23,24,27,13,12,28,32), (4,9)(10,13)(11,22)(12,23)(24,28)(25,26)(27,32)(33,34), (3,4,8,9)(5,13,16,10)(6,12,19,23)(7,11,21,22)(14,26,20,25)(15,28,17,24)(18,27,31,32)(29,34,30,33), (1,2));