C₄×#85544 #72259143

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

solvable

Aliases

direct product

C₄×#85544

Splittings

Fitting

C₄×#585 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄²	GL(2,ℤ/4)
commutator	#41⋊C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×S₄
Fitting	C₄×#585	D₃
socle	C₂²	C₂²×GL(2,ℤ/4)
prime radical	C₂×#85544	C₂
cosocle	C₂×#41⋊C₃	C₂⁴
hypercenter	C₄×#41	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₄×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

3363 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 52 points:

(5,39,44,11,22,41,12,25,37,7,24,40)(6,47,52,17,31,49,18,34,45,13,33,48)(8,27,38,10,26,23,20,19,42,21,9,43)(14,36,46,16,35,32,29,28,50,30,15,51);(5,6,12,18)(7,13,11,17)(8,14,20,29)(9,15,26,35)(10,16,21,30)(19,28,27,36)(22,31,24,33)(23,32,43,51)(25,34,39,47)(37,45,44,52)(38,46,42,50)(40,48,41,49);(7,10)(8,20)(9,23)(11,21)(13,16)(14,29)(15,32)(17,30)(19,37)(22,40)(24,41)(25,38)(26,43)(27,44)(28,45)(31,48)(33,49)(34,46)(35,51)(36,52)(39,42)(47,50);(1,2,3,4)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

C₄×#11923520

GAP

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

gap> G := Group((5,39,44,11,22,41,12,25,37,7,24,40)(6,47,52,17,31,49,18,34,45,13,33,48)(8,27,38,10,26,23,20,19,42,21,9,43)(14,36,46,16,35,32,29,28,50,30,15,51), (5,6,12,18)(7,13,11,17)(8,14,20,29)(9,15,26,35)(10,16,21,30)(19,28,27,36)(22,31,24,33)(23,32,43,51)(25,34,39,47)(37,45,44,52)(38,46,42,50)(40,48,41,49), (7,10)(8,20)(9,23)(11,21)(13,16)(14,29)(15,32)(17,30)(19,37)(22,40)(24,41)(25,38)(26,43)(27,44)(28,45)(31,48)(33,49)(34,46)(35,51)(36,52)(39,42)(47,50), (1,2,3,4));

C4×#85544 #72259143