C₂⁵×#87604 #27158733

order	12416
isoclinism	isoclinic class of #87604
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product	C₂⁵×#87604
unambiguous semidirect product	D₉₇⋊(C₂⁴×C₄)

Splittings

Fitting, socle	C₂⁵×C₉₇ ⋊ C₄
derived, Sylow	C₉₇ ⋊ C₂⁵×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁵	#87604
commutator, hypocenter	C₉₇	C₂⁵×C₄
Frattini	C₁	G
Fitting, socle	C₂⁵×C₉₇	C₄
prime radical	C₂⁵×D₉₇	C₂
cosocle	D₉₇	C₂⁶
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₂⁵

10552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 6 generators on 107 points:

(12,13,14,16)(15,18,19,25)(17,22,23,21)(20,27,28,38)(24,31,32,34)(26,35,36,30)(29,40,41,55)(33,44,45,51)(37,48,49,47)(39,52,53,43)(42,57,58,76)(46,61,62,72)(50,65,66,68)(54,69,70,64)(56,73,74,60)(59,78,79,96)(63,82,83,92)(67,86,87,90)(71,89,84,88)(75,91,80,85)(77,93,94,81)(95,100,101,99)(97,102,103,98)(104,106,107,105);(9,10)(11,12,15,20,29,42,59,80,74,55,72,90,82,57,77,95,102,93,73,52,35,22,13,17,24,33,46,63,84,70,53,38,51,68,88,78,97,104,106,100,91,69,48,31,18,26,37,50,67,87,66,49,36,25,34,47,64,85,99,105,107,103,96,89,65,44,27,39,54,71,83,62,45,32,23,16,21,30,43,60,81,98,101,94,76,92,86,61,40,56,75,79,58,41,28,19,14);(7,8);(5,6);(3,4);(1,2)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21507);

C25×#87604 #27158733