C₂×#11714276 #72266526

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#11714276

Splittings

derived

#5464 ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#85558
commutator, hypocenter	#5464	C₂×C₄
Frattini	C₂³	C₂×#1020
Fitting	C₂²×H(3,𝔽₄)	D₃
socle	C₂⁴	#1020
prime radical	C₂²×#5464	C₂
cosocle	C₂×#5464	C₂²
LCS	G ▷ #5464
UCS	C₁ ◁ C₂²

1528 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 1528 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 28
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×#72210084

GAP

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

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