C₂×#11711647 #72341542

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#11711647

Splittings

Fitting

C₂×#1791 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	#72991
commutator, cosocle	#5950	C₂³
Frattini	C₂²×C₄	C₂²×S₄
Fitting	C₂×#1791	D₃
prime radical	C₂×#87521	C₂
hypercenter	C₂×C₄	#1020⋊C₂
hypocenter	#997	C₂×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1699 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1699 conjugacy classes of subgroups — too many to draw. Only its 43 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236743, C₂×#72236766, C₂×#72236783, …
maximal normal subgroup of	C₂×#72236743, C₂×#72236766, C₂×#72236783, …

GAP

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

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