C₂⁴×(C₂×D₄₇)⋊C₄ #27122829

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂⁴×(C₂×D₄₇)⋊C₄

Splittings

Fitting	C₂⁵×C₄×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂⁴×C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂⁶	C₂×D₄₇
commutator	C₂×C₄₇	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂⁵×C₄×C₄₇	C₂
socle	C₂⁶×C₄₇	C₂²
prime radical	C₂⁶×D₄₇	C₂
cosocle	C₂²×C₄₇	C₂⁶
hypercenter	C₂⁵×C₄	D₄₇
hypocenter	C₄₇	C₂⁴×C₂²⋊C₄
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁶ ◁ C₂⁵×C₄

43678 conjugacy classes of subgroups; only the 10859 normal subgroups are shown.

normal subgroups only

This group has 43678 conjugacy classes of subgroups — too many to draw. Only its 10859 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) = 63
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 6 generators on 63 points:

(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)(56,57,58,59)(61,63);(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)(60,61)(62,63);(54,55);(52,53);(50,51);(48,49)

Conjugacy classes & character table

3104 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1076119);