C₂×#20260760 #27150293

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#20260760

Splittings

Fitting	C₂×C₄⋊C₈×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ C₂×#424

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×C₄⋊C₈×C₉₇	C₂
socle	C₂³×C₉₇	C₂²×C₄
prime radical	C₂³×D₉₇	C₂×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₉₇
hypocenter	C₉₇	C₂×#424
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 4 generators on 119 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)(106,112)(109,115)(113,118)(116,119);(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)(100,101,102,103)(104,106)(105,109)(107,113)(108,112)(110,116)(111,115)(114,118)(117,119);(101,103)(104,105,107,110,108,111,114,117)(106,109,113,116,112,115,118,119);(98,99)

Conjugacy classes & character table

1976 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 13067);

C2×#20260760 #27150293