C₂×#11711644 #72342929

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

Aliases

direct product

C₂×#11711644

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×#87522	C₂
hypercenter	C₂×C₄	#1020⋊C₂
hypocenter	#997	C₂×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1471 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1471 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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236885, C₂×#72236917, C₂×#72236909, …
maximal normal subgroup of	C₂×#72236885, C₂×#72236917, C₂×#72236909, …

GAP

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

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