C₂×(C₂×#11715906)⋊C₄ #27158869

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×(C₂×#11715906)⋊C₄

Splittings

Fitting	C₂²×C₄×C₉₇ ⋊ C₈
Sylow	C₉₇ ⋊ C₂×#336

Subgroups

	subgroup	G/H
center	C₂³	C₂×#11715906
commutator	C₂×C₉₇	C₂×C₄×C₈
Frattini	C₂²	C₂²×#11715906
Fitting	C₂²×C₄×C₉₇	C₈
socle	C₂³×C₉₇	C₂×C₈
prime radical	C₂³×D₉₇	C₂×C₄
cosocle	C₂²×#87604	C₂³
hypercenter	C₂²×C₄	#11715906
hypocenter	C₉₇	C₂×#336
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

528 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 191 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) = 107
transitive degree	μ_t(G) = 1552

A minimal faithful permutation representation — 3 generators on 107 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)(100,101,102,103)(105,107);(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)(104,105)(106,107);(98,99)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21643);