C₂³×#13093940 #27079759

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂³×#13093940

Splittings

Fitting	C₂³×M₁₆×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂³×#100

Subgroups

	subgroup	G/H
center	C₂⁴	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²⋊C₄
prime radical	C₂⁴×D₁₈₈	C₂
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	C₂³×#100
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

7134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 61 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(55,58)(56,60);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,56,59,57,58,60,61);(52,53);(50,51);(48,49)

Conjugacy classes & character table

1928 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033049);

C23×#13093940 #27079759