C₂×#72210002 #72239986

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#72210002

Splittings

Fitting, Sylow	C₂×#6342473 ⋊ C₃
derived	#17338 ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72210002
commutator, cosocle, hypocenter	#17338	C₂²×C₃
Frattini	C₄²	C₂²×C₂²⋊A₄
Fitting	C₂×#6342473	C₃
socle	C₂³	C₂×C₂²⋊(C₄²⋊C₃)
LCS	G ▷ #17338
UCS	C₁ ◁ C₂

5486 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 5486 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 48
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 48 points:

(1,8,11)(2,3,23)(4,26,30)(5,19,41,10,28,46)(6,14,31)(7,15,32)(9,16,43)(12,33,22,24,44,20)(13,34,39,25,45,40)(17,35,29,27,36,42)(18,37,47)(21,38,48);(1,6,2,7)(4,18,9,21)(5,20,10,22)(11,31,23,32)(17,39,27,40)(29,34,42,45)(30,47,43,48)(33,41,44,46);(1,39,7,17,2,40,6,27)(3,38,15,16,8,37,14,26)(4,5,18,20,9,10,21,22)(11,33)(12,36,19,13,24,35,28,25)(23,44)(29,48)(30,34)(31,41)(32,46)(42,47)(43,45)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72239995

GAP

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

gap> G := Group((1,8,11)(2,3,23)(4,26,30)(5,19,41,10,28,46)(6,14,31)(7,15,32)(9,16,43)(12,33,22,24,44,20)(13,34,39,25,45,40)(17,35,29,27,36,42)(18,37,47)(21,38,48), (1,6,2,7)(4,18,9,21)(5,20,10,22)(11,31,23,32)(17,39,27,40)(29,34,42,45)(30,47,43,48)(33,41,44,46), (1,39,7,17,2,40,6,27)(3,38,15,16,8,37,14,26)(4,5,18,20,9,10,21,22)(11,33)(12,36,19,13,24,35,28,25)(23,44)(29,48)(30,34)(31,41)(32,46)(42,47)(43,45));