C₄²×#11715906 #27158868

order	12416
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄³

Aliases

direct product	C₄²×#11715906
unambiguous semidirect product	#87604⋊(C₄×C₈)

Splittings

Fitting	C₄²×C₉₇ ⋊ C₈
derived, Sylow	C₉₇ ⋊ C₄²×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	#11715906
commutator, hypocenter	C₉₇	C₄²×C₈
Frattini	C₂²	C₂²×#11715906
Fitting	C₄²×C₉₇	C₈
socle	C₂²×C₉₇	C₂²×C₈
prime radical	C₂²×D₉₇	C₂²×C₄
cosocle	C₂²×#87604	C₂³
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₄²

408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 105
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 3 generators on 105 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21642);

C42×#11715906 #27158868