C₄×C₈×#87604 #27157387

order	12416
isoclinism	isoclinic class of #87604
automorphism	?
homology	Abelianization (H₁) = C₄²×C₈ · Schur multiplier (H₂) = C₄³

Aliases

direct product

C₄×C₈×#87604

Splittings

Fitting	C₄×C₈×C₉₇ ⋊ C₄
derived, Sylow	C₉₇ ⋊ C₄²×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₈	#87604
commutator, hypocenter	C₉₇	C₄²×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₄×C₈×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₄²
prime radical	C₂²×D₉₇	C₂²×C₄
cosocle	C₂×C₄×D₉₇	C₂³
LCS	G ▷ C₉₇
UCS	C₁ ◁ C₄×C₈

408 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 109
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 3 generators on 109 points:

(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95);(1,2,5,10,19,32,49,70,64,45,62,80,72,47,67,85,92,83,63,42,25,12,3,7,14,23,36,53,74,60,43,28,41,58,78,68,87,94,96,90,81,59,38,21,8,16,27,40,57,77,56,39,26,15,24,37,54,75,89,95,97,93,86,79,55,34,17,29,44,61,73,52,35,22,13,6,11,20,33,50,71,88,91,84,66,82,76,51,30,46,65,69,48,31,18,9,4)(102,103,104,106,105,107,108,109);(98,99,100,101)

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, 20161);