C₂²×#10619530 #72268911

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#10619530

Splittings

Sylow

C₉² ⋊ C₂²×D₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	D₉²
commutator, cosocle	C₂×C₉²	C₂⁴
Frattini	C₂×C₃²	C₂²×D₃²
Fitting	C₂³×C₉²	C₂²
socle	C₂³×C₃²	D₃²
hypocenter	C₉²	C₂²×D₄
LCS	G ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂³

1531 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1531 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 144

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72219950
maximal normal subgroup of	C₂×#72219950

GAP

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

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