C₂²×#15689478 #27085473

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂²×#15689478

Splittings

Fitting	C₂²×C₄×C₈×C₄₇ ⋊ C₂
Sylow	C₄₇ ⋊ C₂²×#488

Subgroups

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂³×D₁₈₈	C₂²
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂²×#488
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

2310 conjugacy classes of subgroups; only the 747 normal subgroups are shown.

normal subgroups only

This group has 2310 conjugacy classes of subgroups — too many to draw. Only its 747 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) = 75
transitive degree	μ_t(G) = 6016

A minimal faithful permutation representation — 5 generators on 75 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)(53,57)(56,59)(60,63,64,72,65,73,71,62)(61,67,68,66,69,70,74,75);(52,54,55,58)(53,56,57,59)(60,62,71,73,65,72,64,63)(61,66,74,67,69,75,68,70);(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)(52,53)(54,56)(55,57)(58,59)(60,61)(62,70)(63,66)(64,74)(65,69)(67,72)(68,71)(73,75);(50,51);(48,49)

Conjugacy classes & character table

3032 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1038763);