C₂×#72210004 #72239991

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂³×C₄

solvable

Aliases

direct product

C₂×#72210004

Splittings

Fitting, Sylow	C₂×#181415 ⋊ C₃
derived	#181415 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72210004
commutator, cosocle, hypocenter	#181415	C₂×C₃
Frattini	C₂×C₄²	C₂×C₂²⋊A₄
Fitting	C₂×#181415	C₃
socle	C₂³	C₂²⋊(#95⋊C₃)
LCS	G ▷ #181415
UCS	C₁ ◁ C₂

3765 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 3765 conjugacy classes of subgroups — too many to draw. Only its 49 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:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72239995, #72240030

GAP

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

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