C₂×#11711640 #72343012

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

Aliases

direct product

C₂×#11711640

Subgroups

	subgroup	G/H
center, socle	C₂²	#73081
commutator, cosocle	#5950	C₂³
Frattini	C₂²×C₄	C₂²×S₄
Fitting	C₂×#3027	D₃
prime radical	C₂×#3027⋊C₃	C₂
hypercenter	C₂×C₄	#1020⋊C₂
hypocenter	#997	C₂×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1467 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1467 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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236908, C₂×#72236886, C₂×#72236914, …
maximal normal subgroup of	C₂×#72236908, C₂×#72236886, C₂×#72236914, …

GAP

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

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