C₂×#11711639 #72342981

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

Aliases

direct product

C₂×#11711639

Splittings

Fitting

C₂×#3028 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	#73081
commutator, cosocle	#5950	C₂³
Frattini	C₂²×C₄	C₂²×S₄
Fitting	C₂×#3028	D₃
prime radical	C₂×#87521	C₂
hypercenter	C₂×C₄	#1020⋊C₂
hypocenter	#997	C₂×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1523 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1523 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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236912, C₂×#72236886, C₂×#72236909, …
maximal normal subgroup of	C₂×#72236912, C₂×#72236886, C₂×#72236909, …

GAP

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

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