(C₂×#11715906)⋊C₈ #27158738

order	12416
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂²×C₄

Aliases

unambiguous semidirect product

(C₂×#11715906)⋊C₈

Splittings

Fitting	C₂×C₈×C₉₇ ⋊ C₈
Sylow	C₉₇ ⋊ #1273

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#11715906
commutator	C₂×C₉₇	C₈²
Fitting	C₂×C₈×C₉₇	C₈
socle	C₂²×C₉₇	C₄×C₈
prime radical	C₂²×D₉₇	C₄²
cosocle	C₂×C₄×#87604	C₂²
hypercenter	C₂×C₈	#11715906
hypocenter	C₉₇	#1273
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

212 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 1552

A minimal faithful permutation representation — 2 generators on 109 points:

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

Polycyclic presentation:

2^7,97|a^2=d,b^2=e,d^2=f,e^2=g,b^a=bc,h^a=h^64,h^d=h^22,h^f=h^96

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21512);