C₂×#11715748 #72344111

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#11715748

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#1020
commutator, cosocle	C₂×#997	C₂³
Frattini	C₂²	C₂²×#1020
Fitting	C₂×C₄×H(5,𝔽₂)	D₃
prime radical	C₂²×#997	C₂²
hypercenter	C₂²×C₄	#1020
hypocenter	#997	C₂×Q₈
LCS	G ▷ C₂×#997 ▷ #997
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

2844 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 2844 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72245747, C₂×#72245595, C₂×#72245729, …
maximal normal subgroup of	C₂×#72245747, C₂×#72245595, C₂×#72245729, …

GAP

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

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