C₂×#72209884 #72228492

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#72209884

Splittings

Fitting	C₂×#15947 ⋊ D₃
derived	#11709313 ⋊ C₂²

Subgroups

	subgroup	G/H
center	C₂²	#11714279
commutator, cosocle, hypocenter	#11709313	C₂²
Frattini	C₂⁴	C₂×#1020
Fitting	C₂×#15947	D₃
socle	C₂⁴	#5934
hypercenter	C₂³	#85559
LCS	G ▷ #11709313
UCS	C₁ ◁ C₂² ◁ C₂³

8478 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 8478 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 26
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#72209884, #72258531, C₂×C₃×#72209884
maximal normal subgroup of	C₄×#72209884, C₂×C₃×#72209884
maximal quotient of	C₄×#72209884, C₂×#72210665, #72255861, …
automorphism group of	C₂×C₃×C₄²
outer automorphism group of	C₂×#95, #376, C₂×C₃×C₄²

GAP

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

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