#72209614

order	5760
isoclinism	isoclinic class of #72209614
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂×C₃×C₄

Aliases

No aliases for this group.

Splittings

Fitting, socle

C₂⁴ ⋊ A₆

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Fitting, socle, cosocle	C₂⁴	A₆
LCS	G
UCS	C₁

139 conjugacy classes of subgroups; only the 3 normal subgroups are shown.

normal subgroups only

This group has 139 conjugacy classes of subgroups — too many to draw. Only its 3 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Permutation generators:

(2,4)(3,5)(7,16)(8,12)(9,14)(10,15);(2,6,7)(3,13,14)(4,15,5)(8,10,11)(9,12,16);(2,8,9)(4,11,10)(5,13,16)(6,14,15);(1,2)(3,10)(4,11)(5,12)(6,7)(8,9)(13,16)(14,15);(1,3)(2,10)(4,8)(5,6)(7,12)(9,11)(13,14)(15,16);(1,4)(2,11)(3,8)(5,15)(6,16)(7,13)(9,10)(12,14);(1,5)(2,12)(3,6)(4,15)(7,10)(8,16)(9,13)(11,14)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	15	160	180	180	360	480	640	720	720	1152	1152
order	1	2	3	2	4	4	6	3	4	8	5	5
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	5	5	2	1	1	1	2	-1	-1	-1	0	0
χ₃	5	5	-1	1	1	1	-1	2	-1	-1	0	0
χ₄	8	8	-1	0	0	0	-1	-1	0	0	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$
χ₅	8	8	-1	0	0	0	-1	-1	0	0	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$
χ₆	9	9	0	1	1	1	0	0	1	1	-1	-1
χ₇	10	10	1	-2	-2	-2	1	1	0	0	0	0
χ₈	15	-1	3	3	-1	-1	-1	0	1	-1	0	0
χ₉	15	-1	3	-1	3	-1	-1	0	-1	1	0	0
χ₁₀	30	-2	-3	2	2	-2	1	0	0	0	0	0
χ₁₁	45	-3	0	1	-3	1	0	0	-1	1	0	0
χ₁₂	45	-3	0	-3	1	1	0	0	1	-1	0	0

Relationships

derived subgroup of	C₂×#72209614, C₂×#72218848, C₂×#72218850, …
maximal subgroup of	#72218848, #72218849, #72218850, …
maximal normal subgroup of	#72218848, #72218849, #72218850, …
maximal quotient of	#72209666, #72209667, #72209668, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((2,4)(3,5)(7,16)(8,12)(9,14)(10,15), (2,6,7)(3,13,14)(4,15,5)(8,10,11)(9,12,16), (2,8,9)(4,11,10)(5,13,16)(6,14,15), (1,2)(3,10)(4,11)(5,12)(6,7)(8,9)(13,16)(14,15), (1,3)(2,10)(4,8)(5,6)(7,12)(9,11)(13,14)(15,16), (1,4)(2,11)(3,8)(5,15)(6,16)(7,13)(9,10)(12,14), (1,5)(2,12)(3,6)(4,15)(7,10)(8,16)(9,13)(11,14));