2₋¹⁺⁴×Dic₉₇ #27154949

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

2₋¹⁺⁴×Dic₉₇

Splittings

Sylow

C₉₇ ⋊ C₄×2₋¹⁺⁴

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₉₇
commutator	C₂×C₉₇	C₂⁴×C₄
Fitting, prime radical	C₂×2₋¹⁺⁴×C₉₇	C₂
socle, cosocle	C₂²×C₉₇	C₂⁵
hypercenter	C₂×2₋¹⁺⁴	D₉₇
hypocenter	C₉₇	C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×2₋¹⁺⁴

1492 conjugacy classes of subgroups; only the 1063 normal subgroups are shown.

normal subgroups only

This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 1063 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) = 117
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 5 generators on 117 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99,100,101);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(103,109)(107,110)(108,117)(115,116);(102,105,106,114)(103,108,109,117)(104,111,112,113)(107,115,110,116);(102,104,106,112)(103,107,109,110)(105,113,114,111)(108,116,117,115);(102,103)(104,110)(105,108)(106,109)(107,112)(111,116)(113,115)(114,117)

Conjugacy classes & character table

1700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 17723);